Mathematics 0- Book Two Trigonometry One Trigonometry Two Permutations and Combinations
A workbook and animated series by Barry Mabillard Copyright 04
This page has been left blank for correct workbook printing.
Mathematics 0- Trigonometry I The Unit Circle Formula Sheet Trigonometry II Note: The unit circle is NOT included on the official formula sheet. Transformations & Operations Exponential and Logarithmic Functions Permutations & Combinations Polynomial, Radical & Rational Functions Curriculum Alignment Math 0-: Alberta Northwest Territories Nunavut Pre-Calculus : British Columbia Yukon Pre-Calculus 0: Saskatchewan Pre-Calculus 40S: Manitoba
Mathematics 0- Table of Contents Unit : Polynomial, Radical, and Rational Functions Lesson : Polynomial Functions Lesson : Polynomial Division Lesson : Polynomial Factoring Lesson 4: Radical Functions Lesson 5: Rational Functions I Lesson 6: Rational Functions II Unit : Transformations and Operations Lesson : Basic Transformations Lesson : Combined Transformations Lesson : Inverses Lesson 4: Function Operations Lesson 5: Function Composition Unit : Exponential and Logarithmic Functions Lesson : Exponential Functions Lesson : Laws of Logarithms Lesson : Logarithmic Functions Unit 4: Trigonometry I Lesson : Degrees and Radians Lesson : The Unit Circle Lesson : Trigonometric Functions I Lesson 4: Trigonometric Functions II Unit 5: Trigonometry II Lesson 5: Trigonometric Equations Lesson 6: Trigonometric Identities I Lesson 7: Trigonometric Identities II Unit 6: Permutations and Combinations Lesson : Permutations Lesson : Combinations Lesson : The Binomial Theorem Total Course 7:45 (6 days) :8 ( days) :9 ( days) : ( days) 0:5 ( days) :00 ( days) : ( days) 4:8 ( days) 0:57 ( days) 0:50 ( days) 0:4 ( days) 0:48 ( days) : ( days) 5:55 ( days) :5 (4 days) : (4 days) :5 (4 days) 9:59 (7 days) : (4 days) :5 (4 days) :4 (5 days) :58 (4 days) 7:05 ( days) : (4 days) :4 (4 days) :9 (4 days) 4:57 (0 days) :00 (4 days) :56 (4 days) :0 ( days) 40:9 (78 days)
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle, θ. b) positive and negative angles: Draw θ = 0 Draw θ = -0 c) reference angle: Find the reference angle of θ = 50.
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 d) co-terminal angles: Draw the first positive co-terminal angle of 60. e) principal angle: Find the principal angle of θ = 40. f) general form of co-terminal angles: Find the first four positive co-terminal angles of θ = 45. Find the first four negative co-terminal angles of θ = 45.
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Three Angle Types: Degrees, Radians, and Revolutions. a) Define degrees, radians, and revolutions. Angle Types and Conversion Multipliers i) Degrees: Draw θ = ii) Radians: Draw θ = rad iii) Revolutions: Draw θ = rev
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 b) Use conversion multipliers to answer the questions and fill in the reference chart. Round all decimals to the nearest hundredth. Conversion Multiplier Reference Chart i) = rad degree radian revolution degree ii) = rev radian iii).6 = revolution iv).6 = rev v) 0.75 rev = vi) 0.75 rev = rad c) Contrast the decimal approximation of a radian with the exact value of a radian. i) Decimal Approximation: 45 = rad ii) Exact Value: 45 = rad
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Convert each angle to the requested form. Round all decimals to the nearest hundredth. a) convert 75 to an approximate radian decimal. Angle Conversion Practice b) convert 0 to an exact-value radian. c) convert 0 to an exact-value revolution. d) convert.5 to degrees. e) convert to degrees. f) write as an approximate radian decimal. g) convert to an exact-value revolution. h) convert 0.5 rev to degrees. i) convert rev to radians.
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 4 The diagram shows commonly used degrees. Find exact-value radians that correspond to each degree. When complete, memorize the diagram. Commonly Used Degrees and Radians a) Method One: Find all exact-value radians using a conversion multiplier. b) Method Two: Use a shortcut. (Counting Radians) 90 = = 50 = 5 = 0 60 = 45 = 0 = 0 = = 80 60 = = 0 = 5 = 40 0 = 5 = 00 = = 70
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 5 a) 0 Draw each of the following angles in standard position. State the reference angle. Reference Angles b) -60 c) 5. d) - 5 4 e) 7
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 6 a) 90 Draw each of the following angles in standard position. State the principal and reference angles. Principal and Reference Angles b) -855 c) 9 d) - 0
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians For each angle, find all co-terminal Example 7 Co-terminal Angles angles within the stated domain. a) 60, Domain: -60 θ < 080 b) -495, Domain: -080 θ < 70 c).78, Domain: - θ < 4 d) 8, Domain: θ < 7 5
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 8 For each angle, use estimation to find the principal angle. a) 89 b) -47.4 Principal Angle of a Large Angle 9 c) d) 5 95 6
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 9 a) principal angle = 00 (find co-terminal angle rotations counter-clockwise) Use the general form of co-terminal angles to find the specified angle. General Form of Co-terminal Angles b) principal angle = 5 (find co-terminal angle 4 rotations clockwise) c) How many rotations are required to find the principal angle of -400? State the principal angle. d) How many rotations are required to find the principal angle of? State the principal angle.
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 0 Six Trigonometric Ratios In addition to the three primary trigonometric ratios (sinθ, cosθ, and tanθ), there are three reciprocal ratios (cscθ, secθ, and cotθ). Given a triangle with side lengths of x and y, and a hypotenuse of length r, the six trigonometric ratios are as follows: sinθ = y r cscθ = sinθ = r y r y cosθ = x r secθ = cosθ = r x θ x tanθ = y x cotθ = tanθ = x y a) If the point P(-5, ) exists on the terminal arm of an angle θ in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle. b) If the point P(, -) exists on the terminal arm of an angle θ in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle.
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Determine the sign of each trigonometric ratio in each quadrant. Signs of Trigonometric Ratios a) sinθ b) cosθ c) tanθ 4 5 5 4 4 5 5 4 4 5 5 4 - θ θ - θ θ - θ θ - θ θ - θ θ - θ θ -4 5 5-4 -4 5 5-4 -4 5 5-4 d) cscθ e) secθ f) cotθ 4 5 5 4 4 5 5 4 4 5 5 4 - θ θ - θ θ - θ θ - θ θ - θ θ - θ θ -4 5 5-4 -4 5 5-4 -4 5 5-4 g) How do the quadrant signs of the reciprocal trigonometric ratios (cscθ, secθ, and cotθ) compare to the quadrant signs of the primary trigonometric ratios (sinθ, cosθ, and tanθ)?
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example Given the following conditions, find the quadrant(s) where the angle θ could potentially exist. What Quadrant(s) is the Angle in? a) i) sinθ < 0 ii) cosθ > 0 iii) tanθ > 0 b) i) sinθ > 0 and cosθ > 0 ii) secθ > 0 and tanθ < 0 iii) cscθ < 0 and cotθ > 0 c) i) sinθ < 0 and cscθ = ii) and cscθ < 0 iii) secθ > 0 and tanθ =
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a radian. Exact Values of Trigonometric Ratios a) b)
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 4 Given one trigonometric ratio, find the exact Exact Values of values of the other five trigonometric ratios. Trigonometric Ratios State the reference angle and the standard position angle, to the nearest hundredth of a degree. a) b)
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 5 Calculating θ with a calculator. Calculator Concerns a) When you solve a trigonometric equation in your calculator, the answer you get for θ can seem unexpected. Complete the following chart to learn how the calculator processes your attempt to solve for θ. If the angle θ could exist in either quadrant or... The calculator always picks quadrant I or II I or III I or IV II or III II or IV III or IV b) Given the point P(-4, ), Mark tries to find the reference angle using a sine ratio, Jordan tries to find it using a cosine ratio, and Dylan tries to find it using a tangent ratio. Why does each person get a different result from their calculator? P(-4, ) Mark s Calculation of θ (using sine) sinθ = 5 Jordan s Calculation of θ (using cosine) cosθ = -4 5 Dylan s Calculation of θ (using tan) tanθ = -4 θ θ = 6.87 θ = 4. θ = -6.87
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 6 Arc Length The formula for arc length is a = rθ, where a is the arc length, θ is the central angle in radians, and r is the radius of the circle. The radius and arc length must have the same units. r a) Derive the formula for arc length, a = rθ. a θ b) Solve for a, to the nearest hundredth. c) Solve for θ. (express your answer as a degree, to the nearest hundredth.) 6 cm 5 cm 5 cm θ a d) Solve for r, to the nearest hundredth. e) Solve for n. (express your answer as an exact-value radian.). cm 5 cm r 6 cm n
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 7 Area of a circle sector. r θ a) Derive the formula for the area of a circle sector, A =. Sector Area r θ In parts (b - e), find the area of each shaded region. b) c) 4 cm 7 6 cm 40 d) e) 9 cm 60 0 6 cm cm
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 8 The formula for angular speed is, where ω (Greek: Omega) is the angular speed, θ is the change in angle, and T is the change in time. Calculate the requested quantity in each scenario. Round all decimals to the nearest hundredth. a) A bicycle wheel makes 00 complete revolutions in minute. Calculate the angular speed in degrees per second. θ b) A Ferris wheel rotates 00 in 4.5 minutes. Calculate the angular speed in radians per second.
0 80 = 7 6 Trigonometry LESSON ONE - Degrees and Radians c) The moon orbits Earth once every 7 days. Calculate the angular speed in revolutions per second. If the average distance from the Earth to the moon is 84 400 km, how far does the moon travel in one second? d) A cooling fan rotates with an angular speed of 400 rpm. What is the speed in rps? e) A bike is ridden at a speed of 0 km/h, and each wheel has a diameter of 68 cm. Calculate the angular speed of one of the bicycle wheels and express the answer using revolutions per second.
Trigonometry LESSON ONE - Degrees and Radians 0 80 = 7 6 Example 9 A satellite orbiting Earth 40 km above the surface makes one complete revolution every 90 minutes. The radius of Earth is approximately 670 km. a) Calculate the angular speed of the satellite. Express your answer as an exact value, in radians/second. 40 km 670 km b) How many kilometres does the satellite travel in one minute? Round your answer to the nearest hundredth of a kilometre.
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x + y = r, where r is the radius of the circle. Draw each circle: i) x + y = 4 ii) x + y = 49 Equation of a Circle 0 0-0 0-0 0-0 -0 b) A circle centered at the origin with a radius of has the equation x + y =. This special circle is called the unit circle. Draw the unit circle and determine if each point exists on the circumference of the unit circle. i) (0.6, 0.8) ii) (0.5, 0.5) - -
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) c) Using the equation of the unit circle, x + y =, find the unknown coordinate of each point. Is there more than one unique answer? i) ii), quadrant II. iii) (-, y) iv), cosθ > 0.
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example The Unit Circle. The Unit Circle The following diagram is called the unit circle. Commonly used angles are shown as radians, and their exact-value coordinates are in brackets. Take a few moments to memorize this diagram. When you are done, use the blank unit circle on the next page to practice drawing the unit circle from memory. questions on next page.
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) a) What are some useful tips to memorize the unit circle? b) Draw the unit circle from memory using a partially completed template. 0
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example Use the unit circle to find the exact value of each trigonometric ratio. Finding Primary Trigonometric Ratios with the Unit Circle a) sin b) cos 80 c) cos 6 4 d) sin 6 e) sin 0 f) cos g) sin 4 h) cos -0 Example 4 Use the unit circle to find the exact value of each trigonometric ratio. a) cos 40 b) -cos c) sin 6 d) cos e) sin 5 9 f) -sin g) cos 4 (-840 ) h) cos 7
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example 5 Other Trigonometric Ratios. Other Trigonometric Ratios The unit circle contains values for cosθ and sinθ only. The other four trigonometric ratios can be obtained using the identities on the right. secθ = cosθ cscθ = sinθ Given angles from the first quadrant of the unit circle, find the exact values of secθ and cscθ. tanθ = sinθ cosθ cotθ = tanθ = cosθ sinθ a) secθ sec = sec = sec = 4 sec = 6 sec 0 = b) cscθ csc = csc = csc = 4 csc = 6 csc 0 =
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example 6 Other Trigonometric Ratios. Other Trigonometric Ratios The unit circle contains values for cosθ and sinθ only. The other four trigonometric ratios can be obtained using the identities on the right. secθ = cosθ cscθ = sinθ Given angles from the first quadrant of the unit circle, find the exact values of tanθ and cotθ. tanθ = sinθ cosθ cotθ = tanθ = cosθ sinθ a) tanθ tan = tan = tan = 4 tan = 6 tan 0 = b) cotθ cot = cot = cot = 4 cot = 6 cot 0 =
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example 7 a) secθ sec Use symmetry to fill in quadrants II, III, and IV for each unit circle. = undefined sec = sec = 4 sec = 6 b) cscθ Symmetry of the Unit Circle csc = 0 csc = csc = 4 csc = 6 sec 0 = csc 0 = undefined c) tanθ tan = undefined tan = tan = 4 tan = 6 d) cotθ cot = 0 cot = cot = 4 cot = 6 tan 0 = 0 cot 0 = undefined
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example 8 Find the exact value of each trigonometric ratio. Finding Reciprocal Trigonometric Ratios with the Unit Circle a) sec 0 b) sec c) csc d) csc 4 e) tan 6 5 f) -tan g) cot 4 (70 ) h) cot 5 6
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example 9 Find the exact value of each trigonometric expression. Evaluating Complex Expressions with the Unit Circle a) b) c) d)
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example 0 Find the exact value of each trigonometric expression. Evaluating Complex Expressions with the Unit Circle a) b) c) d)
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example Find the exact value of each trigonometric ratio. Finding the Trigonometric Ratios of Large Angles with the Unit Circle a) b) c) d)
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example Verify each trigonometric statement with a calculator. Note: Every question in this example has already been seen earlier in the lesson. Evaluating Trigonometric Ratios with a Calculator a) b) c) d) e) f) g) h)
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example Answer each of the following questions related to the unit circle. Coordinate Relationships on the Unit Circle a) What is meant when you are asked to find on the unit circle? b) Find one positive and one negative angle such that P(θ) = c) How does a half-rotation around the unit circle change the coordinates? If θ =, find the coordinates of the point halfway around the unit circle. 6 d) How does a quarter-rotation around the unit circle change the coordinates? If θ =, find the coordinates of the point a quarter-revolution (clockwise) around the unit circle. e) What are the coordinates of P()? Express coordinates to four decimal places.
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example 4 Answer each of the following questions related to the unit circle. a) What is the circumference of the unit circle? Circumference and Arc Length of the Unit Circle b) How is the central angle of the unit circle related to its corresponding arc length? c) If a point on the terminal arm rotates from P(θ) = (, 0) to P(θ) =, what is the arc length? d) What is the arc length from point A to point B on the unit circle? θ A θ B θ
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example 5 Answer each of the following questions related to the unit circle. a) Is sinθ = possible? Explain, using the unit circle as a reference. Domain and Range of the Unit Circle b) Which trigonometric ratios are restricted to a range of - y? Which trigonometric ratios exist outside that range? Range Number Line cosθ & sinθ cscθ & secθ tanθ & cotθ
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle c) If exists on the unit circle, how can the unit circle be used to find cosθ? How many values for cosθ are possible? d) If exists on the unit circle, how can the equation of the unit circle be used to find sinθ? How many values for sinθ are possible? e) If cosθ = 0, and 0 θ <, how many values for sinθ are possible?
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example 6 Complete the following questions related to the unit circle. Unit Circle Proofs a) Use the Pythagorean Theorem to prove that the equation of the unit circle is x + y =. b) Prove that the point where the terminal arm intersects the unit circle, P(θ), has coordinates of (cosθ, sinθ). c) If the point θ exists on the terminal arm of a unit circle, find the exact values of the six trigonometric ratios. State the reference angle and standard position angle to the nearest hundredth of a degree.
(cosθ, sinθ) Trigonometry LESSON TWO - The Unit Circle Example 7 In a video game, the graphic of a butterfly needs to be rotated. To make the butterfly graphic rotate, the programmer uses the equations: x = x cos θ - y sin θ y = x sin θ + y cos θ to transform each pixel of the graphic from its original coordinates, (x, y), to its new coordinates, (x, y ). Pixels may have positive or negative coordinates. a) If a particular pixel with coordinates of (50, 00) is rotated by, what are the new 6 coordinates? Round coordinates to the nearest whole pixel. 5 b) If a particular pixel has the coordinates (640, 480) after a rotation of, what were the 4 original coordinates? Round coordinates to the nearest whole pixel.
Trigonometry LESSON TWO - The Unit Circle (cosθ, sinθ) Example 8 From the observation deck of the Calgary Tower, an observer has to tilt their head θ A down to see point A, and θ B down to see point B. a) Show that the height of the observation x deck is h =. cotθ A - cotθ B h θ B θ A A B x b) If θ A =, θ B =, and x =.9 m, how high is the observation deck above the ground, to the nearest metre?
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I a) Example Label all tick marks in the following grids and state the coordinates of each point. y Trigonometric Coordinate Grids 5 0 θ -5 b) y 0 0 θ -0
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d c) y 8 0 8 θ - d) y 40 4 0 4 θ -40
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example a) Draw y = sinθ. Exploring the graph of y = sinθ. y y = sinθ 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0 7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ - b) State the amplitude. c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. Unit Circle Reference f) State the θ-intercepts. Write your answer using a general form expression. g) State the y-intercept. h) State the domain and range.
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example a) Draw y = cosθ. Exploring the graph of y = cosθ. y y = cosθ 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0 7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ - b) State the amplitude. c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. Unit Circle Reference f) State the θ-intercepts. Write your answer using a general form expression. g) State the y-intercept. h) State the domain and range.
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example 4 a) Draw y = tanθ. Exploring the graph of y = tanθ. y y = tanθ 7 5 4 5 7 5 5 7 5 4 5 7 6 4 4 6 6 4 4 6 6 4 4 6 6 4 4 6 θ - - b) Is it correct to say a tangent graph has an amplitude? c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. f) State the θ-intercepts. Write your answer using a general form expression. tan = - tan = - 4 5 tan = 6 - Unit Circle Reference tan = undefined tan = tan = 4 tan = 6 g) State the y-intercept. h) State the domain and range. tan = 0 7 tan = 6 5 tan = 4 4 tan = tan tan 0 = 0 tan = - 6 7 tan = - 4 5 tan = - = undefined
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example 5 Graph each function over the domain 0 θ. The base graph is provided as a convenience. a) y = sinθ b) y = -cosθ The a Parameter 5 5 0 0-5 -5 c) y = sinθ 5 d) y = cosθ 5 5 0 0-5 -5
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Determine the trigonometric function corresponding to each graph. a) write a sine function. b) write a sine function. 8 Example 6 8 The a Parameter 0 0-8 -8 c) write a cosine function. d) write a cosine function. 5 0 0 ( ), 4 - -5
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d 5 Example 7 a) y = sinθ - Graph each function over the domain 0 θ. The base graph is provided as a convenience. b) y = cosθ + 4 5 The d Parameter 0 0-5 -5 c) y = - sinθ + d) y = cosθ - 5 5 0 0-5 -5
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I a) write a sine function. 4 Example 8 Determine the trigonometric function corresponding to each graph. b) write a cosine function. 5 The d Parameter 0 0-4 -5 c) write a cosine function. d) write a sine function. 4 0 0 - -4
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example 9 Graph each function over the stated domain. The base graph is provided as a convenience. a) y = cosθ (0 θ ) b) y = sinθ (0 θ ) The b Parameter 0 0 - - c) y = cos θ (0 θ 6) d) y = sin θ (0 θ 0) 5 0 4 6 0 4 6 8 0 - -
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example 0 Graph each function over the stated domain. The base graph is provided as a convenience. a) y = -sin(θ) (- θ ) b) y = 4cosθ + 6 (- θ ) The b Parameter 6 - - - - -6 - - c) y = cos θ - (- θ ) 4 d) y = sin θ (0 θ 6) - - 0 4 6 - - - -
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example a) write a cosine function. Determine the trigonometric function corresponding to each graph. The b Parameter 0 - b) write a cosine function. 4 0 6-4
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I c) write a sine function. The b Parameter 0 6 9 - d) write a sine function. 0 -
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example Graph each function over the stated domain. The base graph is provided as a convenience. The c Parameter a) (-4 θ 4) b) (-4 θ 4) -4-4 -4-4 - - c) (- θ ) d) (- θ ) 4 - - - - - -4
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example Graph each function over the stated domain. The base graph is provided as a convenience. The c Parameter a) θ b) (- θ 6) - 4 6 - - c) (- θ 4) d) (- θ ) 4-4 - - - - -4
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example 4 a) write a cosine function. Determine the trigonometric function corresponding to each graph. The c Parameter - b) write a sine function. 6 0 4-6
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I c) write a sine function. The c Parameter - - d) write a cosine function. 4-8 -4 4 8-4
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example 5 Graph each function over the stated domain. The base graph is provided as a convenience. a, b, c, & d a) (0 θ 6) b) (0 θ ) 5 0 4 6 0-5 - c) - (0 θ ) d) (0 θ ) 5 6 0 0-5 -6
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example 6 Write a trigonometric function for each graph. a, b, c, & d a) - - 4 - b) - - -
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example 7 a) Draw y = secθ. Exploring the graph of y = secθ. y Graphing Reciprocal Functions 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0-7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ - - b) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. Unit Circle Reference (for secθ) e) Given the graph of f(θ) = cosθ, draw y =. f(θ) y sec = - sec = - 4 5 sec = 6 - sec = undefined sec = sec = 4 sec = 6 sec = - sec 0 = - 0 θ 7 sec = - 6 5 sec = - 4 4 sec = - sec sec = 6 7 sec = 4 5 sec = = undefined
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example 8 a) Draw y = cscθ. Exploring the graph of y = cscθ. y Graphing Reciprocal Functions 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0-7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ - - b) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. Unit Circle Reference (for cscθ) e) Given the graph of f(θ) = sinθ, draw y =. f(θ) y csc = csc = 4 5 csc = 6 csc = csc = csc = 4 csc = 6 csc = undefined csc 0 = undefined - 0 θ 7 csc = - 6 5 csc = - 4 4 csc = - csc = - csc = - 6 7 csc = - 4 5 csc = -
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d Example 9 a) Draw y = cotθ. Exploring the graph of y = cotθ. y Graphing Reciprocal Functions 7 5 4 5 7 5 5 7 5 4 5 7 6 4 4 6 6 4 4 6 6 4 4 6 6 4 4 6 θ - - b) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. Unit Circle Reference (for cotθ) e) Given the graph of f(θ) = tanθ, draw y =. f(θ) y cot = - cot = - 4 5 cot = 6 - cot = 0 cot = cot = 4 cot = 6 cot = undefined cot 0 = undefined - 0 θ 7 cot = 6 5 cot = 4 4 cot = cot = 0 cot = - 6 7 cot = - 4 5 cot = -
y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example 0 Graph each function over the domain 0 θ. The base graph is provided as a convenience. State the new domain and range. Transformations of Reciprocal Functions a) b) 0 0 - y = secθ - y = secθ c) d) 0 0 - y = cscθ - y = cotθ
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d This page has been left blank for correct workbook printing.
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 Trigonometric Functions of Angles Trigonometric Functions of Angles a) b) i) Graph: (0 θ < ) i) Graph: (0º θ < 540º) y y θ 80º 60º 540º θ - y = cosθ (one cycle shown) - y = cosθ (one cycle shown) - - ii) Graph this function using technology. ii) Graph this function using technology.
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t a) h Example 7 i) Graph: Trigonometric Functions of Real Numbers. b) i) Graph: y Trigonometric Functions of Real Numbers 5 0 45 60 t 8 6 4 x - - - - ii) Graph this function using technology. ii) Graph this function using technology. c) What are three differences between trigonometric functions of angles and trigonometric functions of real numbers?
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 Determine the view window for each function and sketch each graph. Graph Preperation and View Windows a) b)
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 4 Determine the view window for each function and sketch each graph. Graph Preperation and View Windows a) b)
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 5 a) write a cosine function. Determine the trigonometric function corresponding to each graph. Find the Trigonometric Function of a Graph 0 0 8 6-0 b) write a sine function. 5-4 8 6-5
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t c) write a cosine function. 0 (8, 9) 0 5 (6, -) -0 d) write a sine function. 00 (45, 50) 0 400 (00, -50) -00
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 6 Answer the following questions: Assorted Questions a) If the transformation g(θ) - = f(θ) is applied to the graph of f(θ) = sinθ, find the new range. b) Find the range of 4. c) If the range of y = cosθ + d is [-4, k], determine the values of d and k.
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t d) State the range of f(θ) - = msin(θ) + n. e) The graphs of f(θ) and g(θ) intersect at the points and If the amplitude of each graph is quadrupled, determine the new points of intersection.
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 77 Answer the following questions: Assorted Questions a) If the point lies on the graph of, find the value of a. b) Find the y-intercept of.
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t c) The graphs of f(θ) and g(θ) intersect at the point (m, n). Find the value of f(m) + g(m). (m, n) g(θ) n f(θ) m d) The graph of f(θ) = kcosθ is transformed to the graph of g(θ) = bcosθ by a vertical stretch about the x-axis. If the point exists on the graph of g(θ), state the vertical stretch factor. k b f(θ) g(θ)
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 8 The graph shows the height of a pendulum bob as a function of time. One cycle of a pendulum consists of two swings - a right swing and a left swing. h(t) cm 8 cm 4 cm ground level 0 cm s s s 4 s t a) Write a function that describes the height of the pendulum bob as a function of time. b) If the period of the pendulum is halved, how will this change the parameters in the function you wrote in part (a)? c) If the pendulum is lowered so its lowest point is cm above the ground, how will this change the parameters in the function you wrote in part (a)?
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 9 A wind turbine has blades that are 0 m long. An observer notes that one blade makes complete rotations (clockwise) every minute. The highest point of the blade during the rotation is 05 m. a) Using Point A as the starting point of the graph, draw the height of the blade over two rotations. A h(t) t b) Write a function that corresponds to the graph. c) Do we get a different graph if the wind turbine rotates counterclockwise?
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 0 A person is watching a helicopter ascend from a distance 50 m away from the takeoff point. a) Write a function, h(θ), that expresses the height as a function of the angle of elevation. Assume the height of the person is negligible. h θ 50 m b) Draw the graph, using an appropriate domain. h(θ) θ c) Explain how the shape of the graph relates to the motion of the helicopter.
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 A mass is attached to a spring 4 m above the ground and allowed to oscillate from its equilibrium position. The lowest position of the mass is.8 m above the ground, and it takes s for one complete oscillation. a) Draw the graph for two full oscillations of the mass. h(t) t b) Write a sine function that gives the height of the mass above the ground as a function of time. c) Calculate the height of the mass after. seconds. Round your answer to the nearest hundredth. d) In one oscillation, how many seconds is the mass lower than. m? Round your answer to the nearest hundredth.
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 A Ferris wheel with a radius of 5 m rotates once every 00 seconds. Riders board the Ferris wheel using a platform m above the ground. a) Draw the graph for two full rotations of the Ferris wheel. h(t) t b) Write a cosine function that gives the height of the rider as a function of time. c) Calculate the height of the rider after.6 rotations of the Ferris wheel. Round your answer to the nearest hundredth. d) In one rotation, how many seconds is the rider higher than 6 m? Round your answer to the nearest hundredth.
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 The following table shows the number of daylight hours in Grande Prairie. December March June September December 6h, 46m h, 7m 7h, 49m h, 7m 6h, 46m a) Convert each date and time to a number that can be used for graphing. Day Number December = March = June = September = December = Daylight Hours 6h, 46m = h, 7m = 7h, 49m = h, 7m = h, 46m = b) Draw the graph for one complete cycle (winter solstice to winter solstice). d(n) n
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II c) Write a cosine function that relates the number of daylight hours, d, to the day number, n. d) How many daylight hours are there on May? Round your answer to the nearest hundredth. e) In one year, approximately how many days have more than 7 daylight hours? Round your answer to the nearest day.
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 4 The highest tides in the world occur between New Brunswick and Nova Scotia, in the Bay of Fundy. Each day, there are two low tides and two high tides. The chart below contains tidal height data that was collected over a 4-hour period. Bay of Fundy Low Tide Time Decimal Hour Height of Water (m) : AM.48 Note: Actual tides at the Bay of Fundy are 6 hours and minutes apart due to daily changes in the position of the moon. High Tide 8: AM. In this example, we will use 6 hours for simplicity. Low Tide : PM.48 High Tide 8: PM. a) Convert each time to a decimal hour. b) Graph the height of the tide for one full cycle (low tide to low tide). h(t) t
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II c) Write a cosine function that relates the height of the water to the elapsed time. d) What is the height of the water at 6:09 AM? Round your answer to the nearest hundredth. e) For what percentage of the day is the height of the water greater than m? Round your answer to the nearest tenth.
Mice Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 5 A wooded region has an ecosystem that supports both owls and mice. Owl and mice populations vary over time according to the equations: Owl population: Mouse population: where O is the population of owls, M is the population of mice, and t is the time in years. a) Graph the population of owls and mice over six years. Population 6000 000 8000 00 Owls 50 00 0 4 5 6 7 8 Time (years) b) Describe how the graph shows the relationship between owl and mouse populations.
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 6 The angle of elevation between the 6:00 position and the :00 position of a historical building s clock, as measured from an observer standing on a hill, is 444. The observer also knows that he is standing 44 m away from the clock, and his eyes are at the same height as the base of the clock. The radius of the clock is the same as the length of the minute hand. If the height of the minute hand s tip is measured relative to the bottom of the clock, what is the height of the tip at 5:08, to the nearest tenth of a metre? 444 44 m
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 Shane is on a Ferris wheel, and his height can be described by the equation. Tim, a baseball player, can throw a baseball with a speed of 0 m/s. If Tim throws a ball directly upwards, the height can be determined by the equation h ball (t) = -4.905t + 0t + If Tim throws the baseball 5 seconds after the ride begins, when are Shane and the ball at the same height?
Trigonometry LESSON FIVE - Trigonometric Equations Example Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Primary Ratios Solving equations with the unit circle. a) b) c) 0 d) tan θ =
Trigonometry LESSON FIVE - Trigonometric Equations a) Example sinθ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. b) sinθ = - Primary Ratios Solving equations graphically with intersection points - - c) cosθ d) cosθ = - - - e) tanθ f) tanθ = undefined - - - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example Find all angles in the domain 0 θ 60 that satisfy the given equation. Write the general solution. Primary Ratios Solving equations with a calculator. (degree mode) a) 90 80 0 60 70 b) 90 80 0 60 c) 70 90 80 0 60 70
Trigonometry LESSON FIVE - Trigonometric Equations Example 4 a) sinθ = Find all angles in the domain 0 θ that satisfy the given equation. Intersection Point(s) of Original Equation Primary Ratios Solving equations graphically with θ-intercepts. θ-intercepts - - - - - - b) cosθ = Intersection Point(s) of Original Equation θ-intercepts - - - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example 5 Solve a) non-graphically, using the cos - feature of a calculator. - 0 θ Primary Ratios Equations with b) non-graphically, using primary trig ratios the unit circle. c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts. - - - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example 6 a) non-graphically, using the sin - feature of a calculator. Solve sinθ = -0.0 θ ε R Primary Ratios Equations with primary trig ratios b) non-graphically, using the unit circle. c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts. - - - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example 7 Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Reciprocal Ratios Solving equations with the unit circle. a) b) c)
Trigonometry LESSON FIVE - Trigonometric Equations Example 8 a) θ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. b) θ Reciprocal Ratios Solving equations graphically with intersection points - - - - c) θ d) secθ = - - - - - e) θ f) θ - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example 9 Find all angles in the domain 0 θ 60 that satisfy the given equation. Write the general solution Reciprocal Ratios Solving equations with a calculator. (degree mode) a) 90 80 0 60 70 b) 90 80 0 60 c) 70 90 80 0 60 70
Trigonometry LESSON FIVE - Trigonometric Equations Example 0 a) θ Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. Intersection Point(s) of Original Equation Reciprocal Ratios Solving equations graphically with θ-intercepts. θ-intercepts - - - - - - b) θ Intersection Point(s) of Original Equation θ-intercepts - - - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example a) non-graphically, using the sin - feature of a calculator. Solve cscθ = - 0 θ Reciprocal Ratios b) non-graphically, using the unit circle. Equations with reciprocal trig ratios c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts. - - - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example a) non-graphically, using the cos - feature of a calculator. Solve secθ = -.66 0 θ 60 b) non-graphically, using the unit circle. Reciprocal Ratios Equations with reciprocal trig ratios c) graphically, using the point(s) of intersection. d) graphically, using θ-intercepts. - - - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example Find all angles in the domain 0 θ that satisfy the given equation. Write the general solution. a) cosθ - = 0 b) θ First-Degree Trigonometric Equations c) tanθ - 5 = 0 d) 4secθ + = secθ +
Trigonometry LESSON FIVE - Trigonometric Equations Example 4 Find all angles in the domain 0 θ that satisfy the given equation. a) sinθcosθ = cosθ b) 7sinθ = 4sinθ First-Degree Trigonometric Equations Check the solution graphically. Check the solution graphically. - - c) sinθtanθ = sinθ d) tanθ + cosθtanθ = 0 Check the solution graphically. Check the solution graphically. - -
Trigonometry LESSON FIVE - Trigonometric Equations Example 5 Find all angles in the domain 0 θ that satisfy the given equation. a) sin θ = b) 4cos θ - = 0 Second-Degree Trigonometric Equations Check the solution graphically. Check the solution graphically. - - c) cos θ = cosθ d) tan 4 θ - tan θ = 0 Check the solution graphically. Check the solution graphically. - - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example 6 a) sin θ - sinθ - = 0 Find all angles in the domain 0 θ that satisfy the given equation. Second-Degree Trigonometric Equations Check the solution graphically. - - - b) csc θ - cscθ + = 0 Check the solution graphically. - - - c) sin θ - 5sin θ + sinθ = 0 Check the solution graphically. - - -
Trigonometry LESSON FIVE - Trigonometric Equations Example 7 Solve each trigonometric equation. Double and Triple Angles a) θ 0 θ i) graphically: ii) non-graphically: - b) θ 0 θ i) graphically: ii) non-graphically: -
Trigonometry LESSON FIVE - Trigonometric Equations Example 8 Solve each trigonometric equation. Half and Quarter Angles a) θ 0 θ 4 i) graphically: ii) non-graphically: 4 - b) θ - 0 θ 8 i) graphically: ii) non-graphically: 4 6 8 -
Trigonometry LESSON FIVE - Trigonometric Equations Example 9 It takes the moon approximately 8 days to go through all of its phases. New Moon First Quarter Full Moon Last Quarter New Moon a) Write a function, P(t), that expresses the visible percentage of the moon as a function of time. Draw the graph. Visible %.00 0.50 0 7 4 8 t b) In one cycle, for how many days is 60% or more of the moon s surface visible?
Trigonometry LESSON FIVE - Trigonometric Equations Example 0 Rotating Sprinkler N A rotating sprinkler is positioned 4 m away from the wall of a house. The wall is 8 m long. As the sprinkler rotates, the stream of water splashes the house d meters from point P. Note: North of point P is a positive distance, and south of point P is a negative distance. a) Write a tangent function, d(θ), that expresses the distance where the water splashes the wall as a function of the rotation angle θ. W S E θ P d b) Graph the function for one complete rotation of the sprinkler. Draw only the portion of the graph that actually corresponds to the wall being splashed. 8 4 d -4 θ -8 c) If the water splashes the wall.0 m north of point P, what is the angle of rotation (in degrees)?
Trigonometry LESSON FIVE - Trigonometric Equations Example Inverse Trigonometric Functions When we solve a trigonometric equation like cosx = -, one possible way to write the solution is: Inverse Trigonometric Functions Enrichment Example Students who plan on taking university calculus should complete this example. In this example, we will explore the inverse functions of sine and cosine to learn why taking an inverse actually yields the solution. a) When we draw the inverse of trigonometric graphs, it is helpful to use a grid that is labeled with both radians and integers. Briefly explain how this is helpful. y 6 5 4-6 -5-4 - - - 4 5 6 0 - - x - -4-5 -6
Trigonometry LESSON FIVE - Trigonometric Equations b) Draw the inverse function of each graph. State the domain and range of the original and inverse graphs (after restricting the domain of the original so the inverse is a function). y = sinx y 6 y = cosx y 6 5 4 5 4-6 -5-4 - - - 0 - - 4 5 6 x -6-5 -4 - - - 0 - - 4 5 6 x - - -4-5 -4-5 -6-6 c) Is there more than one way to restrict the domain of the original graph so the inverse is a function? If there is, generalize the rule in a sentence. d) Using the inverse graphs from part (b), evaluate each of the following:
Trigonometry LESSON SIX - Trigonometric Identities I Example Understanding Trigonometric Identities. a) Why are trigonometric identities considered to be a special type of trigonometric equation? Trigonometric Identities A trigonometric equation that IS an identity: A trigonometric equation that is NOT an identity: - - - - - - b) Which of the following trigonometric equations are also trigonometric identities? i) ii) iii) - - - - - - - iv) v) - - - - - -
Trigonometry LESSON SIX- Trigonometric Identities I Example The Pythagorean Identities. a) Using the definition of the unit circle, derive the identity sin x + cos x =. Why is sin x + cos x = called a Pythagorean Identity? Pythagorean Identities b) Verify that sin x + cos x = is an identity using i) x = and ii) x =. c) Verify that sin x + cos x = is an identity using a graphing calculator to draw the graph. sin x + cos x = -
Trigonometry LESSON SIX - Trigonometric Identities I d) Using the identity sin x + cos x =, derive + cot x = csc x and tan x + = sec x. e) Verify that + cot x = csc x and tan x + = sec x are identities for x =. f) Verify that + cot x = csc x and tan x + = sec x are identities graphically. + cot x = csc x tan x + = sec x - - - - - -
Trigonometry LESSON SIX- Trigonometric Identities I a) Example Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Reciprocal Identities NOTE: You will need to use a graphing calculator to obtain the graphs in this lesson. Make sure the calculator is in RADIAN mode, and use window settings that match the grid provided in each example. Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - - b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -
Trigonometry LESSON SIX - Trigonometric Identities I a) Example 4 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Reciprocal Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - - b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - -
Trigonometry LESSON SIX- Trigonometric Identities I a) Example 5 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -
Trigonometry LESSON SIX - Trigonometric Identities I c) Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - d) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -
Trigonometry LESSON SIX- Trigonometric Identities I a) Example 6 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -
Trigonometry LESSON SIX - Trigonometric Identities I c) Pythagorean Identities Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) d) 0 Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) -
Trigonometry LESSON SIX- Trigonometric Identities I a) Example 7 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - - b) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - -
Trigonometry LESSON SIX - Trigonometric Identities I c) Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - - d) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - -
Trigonometry LESSON SIX- Trigonometric Identities I a) Example 8 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - b) - Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - -
Trigonometry LESSON SIX - Trigonometric Identities I c) Common Denominator Proofs Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - - d) Rewrite the identity so it is absolutely true. (i.e. Include restrictions on the variable) - - -
Trigonometry LESSON SIX- Trigonometric Identities I Example 9 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)
Trigonometry LESSON SIX - Trigonometric Identities I Example 0 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)
Trigonometry LESSON SIX- Trigonometric Identities I Example Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)
Trigonometry LESSON SIX - Trigonometric Identities I Example Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. c) State the non-permissible values for. d) Show graphically that Are the graphs exactly the same? y = sinx - y = tanxcosx -
Trigonometry LESSON SIX- Trigonometric Identities I Example Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. c) State the non-permissible values d) Show graphically that for. Are the graphs exactly the same? y = - - - y = - - -
Trigonometry LESSON SIX - Trigonometric Identities I Example 4 Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. d) Show graphically that c) State the the non-permissible values for. Are the graphs exactly the same? y = - - - y = - - -
Trigonometry LESSON SIX- Trigonometric Identities I Example 5 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) - - - - - - c) d) 6 4 - -4-6 -
Trigonometry LESSON SIX - Trigonometric Identities I Example 6 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) 0 - - - -0 c) d) - -
Trigonometry LESSON SIX- Trigonometric Identities I Example 7 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) 0 0 - -6-9 -0 - c) d) - - - -
Trigonometry LESSON SIX - Trigonometric Identities I Example 8 Use the Pythagorean identities to find the indicated value and draw the corresponding triangle. Pythagorean Identities and Finding an Unknown a) If the value of find the value of cosx within the same domain. b) If the value of, find the value of seca within the same domain. 7 c) If cosθ =, and cotθ < 0, find the exact value of sinθ. 7
Trigonometry LESSON SIX- Trigonometric Identities I Example 9 Trigonometric Substitution. Trigonometric Substitution a) Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to b. θ a b b) Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to a. θ a 4
Trigonometry LESSON SEVEN - Trigonometric Identities II Example Evaluate each trigonometric sum or difference. Sum and Difference Identities a) b) c) d) e) f)
Trigonometry LESSON SEVEN- Trigonometric Identities II Example Write each expression as a single trigonometric ratio. Sum and Difference Identities a) b) c)
Trigonometry LESSON SEVEN - Trigonometric Identities II a) Example Find the exact value of each expression. Sum and Difference Identities b) c) d) Given the exact values of cosine and sine for 5, fill in the blanks for the other angles.
Trigonometry LESSON SEVEN- Trigonometric Identities II a) Example 4 Find the exact value of each expression. For simplicity, do not rationalize the denominator. Sum and Difference Identities b) c)
Trigonometry LESSON SEVEN - Trigonometric Identities II Example 5 Double-angle identities. Double-Angle Identities a) Prove the double-angle sine identity, sinx = sinxcosx. b) Prove the double-angle cosine identity, cosx = cos x - sin x. c) The double-angle cosine identity, cosx = cos x - sin x, can be expressed as cosx = - sin x or cosx = cos x -. Derive each identity. d) Derive the double-angle tan identity,.
Trigonometry LESSON SEVEN- Trigonometric Identities II Example 6 Double-angle identities. Double-Angle Identities a) Evaluate each of the following expressions using a double-angle identity. b) Express each of the following expressions using a double-angle identity. c) Write each of the following expression as a single trigonometric ratio using a double-angle identity.
Trigonometry LESSON SEVEN - Trigonometric Identities II Example 7 Prove each trigonometric identity. Note: Variable restrictions may be ignored for the proofs in this lesson. Sum and Difference Identities a) b) c) d)
Trigonometry LESSON SEVEN- Trigonometric Identities II Example 8 Prove each trigonometric identity. Sum and Difference Identities a) b) c) d)
Trigonometry LESSON SEVEN - Trigonometric Identities II Example 9 Prove each trigonometric identity. Double-Angle Identities a) b) c) d)
Trigonometry LESSON SEVEN- Trigonometric Identities II Example 0 Prove each trigonometric identity. Double-Angle Identities a) b) c) d)
Trigonometry LESSON SEVEN - Trigonometric Identities II Example Prove each trigonometric identity. Assorted Proofs a) b) c) d)
Trigonometry LESSON SEVEN- Trigonometric Identities II Example Prove each trigonometric identity. Assorted Proofs a) b) c) d)
Trigonometry LESSON SEVEN - Trigonometric Identities II Example Prove each trigonometric identity. Assorted Proofs a) b) c) d)
Trigonometry LESSON SEVEN- Trigonometric Identities II Example 4 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d)
Trigonometry LESSON SEVEN - Trigonometric Identities II Example 5 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d)
Trigonometry LESSON SEVEN- Trigonometric Identities II Example 6 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d) 0
Trigonometry LESSON SEVEN - Trigonometric Identities II Example 7 Solve each trigonometric equation over the domain 0 x. Assorted Equations a) b) c) d)
Trigonometry LESSON SEVEN- Trigonometric Identities II Example 8 Trigonometric identities and geometry. A a) Show that B C b) If A = and B = 89, what is the value of C?
Trigonometry LESSON SEVEN - Trigonometric Identities II Example 9 Trigonometric identities and geometry. Solve for x. Round your answer to the nearest tenth. x 57 76 04 B 5 A
Trigonometry LESSON SEVEN- Trigonometric Identities II Example 0 If a cannon shoots a cannonball θ degrees above the horizontal, the horizontal distance traveled by the cannonball before it hits the ground can be found with the function: θ d ( ) θ = v i sinθ cosθ 4.9 The initial velocity of the cannonball is 6 m/s. a) Rewrite the function so it involves a single trigonometric identity. b) Graph the function. Use the graph to describe the trajectory of the cannonball at the following angles: 0, 45, and 90. d θ c) If the cannonball travels a horizontal distance of 00 m, find the angle of the cannon. Solve graphically, and round your answer to the nearest tenth of a degree.
Trigonometry LESSON SEVEN - Trigonometric Identities II Example An engineer is planning the construction of a road through a tunnel. In one possible design, the width of the road maximizes the area of a rectangle inscribed within the cross-section of the tunnel. The angle of elevation from the centre line of the road to the upper corner of the rectangle is θ. Sidewalks on either side of the road are included in the design. a) If the area of the rectangle can be represented by the function A(θ) = msinθ, what is the value of m? sidewalk θ 70 m road width sidewalk b) What angle maximizes the area of the rectangular cross-section? c) For the angle that maximizes the area: i) What is the width of the road? ii) What is the height of the tallest vehicle that will pass through the tunnel? iii) What is the width of one of the sidewalks? Express answers as exact values. A θ
Trigonometry LESSON SEVEN- Trigonometric Identities II Example The improper placement of speakers for a home theater system may result in a diminished sound quality at the primary viewing area. This phenomenon occurs because sound waves interact with each other in a process called interference. When two sound waves undergo interference, they combine to form a resultant sound wave that has an amplitude equal to the sum of the component sound wave amplitudes. If the amplitude of the resultant wave is larger than the component wave amplitudes, we say the component waves experienced constructive interference. If the amplitude of the resultant wave is smaller than the component wave amplitudes, we say the component waves experienced destructive interference. a) Two sound waves are represented with f(θ) and g(θ). i) Draw the graph of y = f(θ) + g(θ) and determine the resultant wave function. ii) Is this constructive or destructive interference? iii) Will the new sound be louder or quieter than the original sound? 6 g(θ) = 4cosθ f(θ) = cosθ 0-6
Trigonometry LESSON SEVEN - Trigonometric Identities II b) A different set of sound waves are represented with m(θ) and n(θ). i) Draw the graph of y = m(θ) + n(θ) and determine the resultant wave function. ii) Is this constructive or destructive interference? iii) Will the new sound be louder or quieter than the original sound? 6 m(θ) = cosθ 0 n(θ) = cos(θ - ) -6 c) Two sound waves experience total destructive interference if the sum of their wave functions is zero. Given p(θ) = sin(θ - /4) and q(θ) = sin(θ - 7/4), show that these waves experience total destructive interference.
Trigonometry LESSON SEVEN- Trigonometric Identities II Example Even & Odd Identities Even & Odd Identities a) Explain what is meant by the terms even function and odd function. b) Explain how the even & odd identities work. (Reference the unit circle or trigonometric graphs in your answer.) c) Prove the three even & odd identities algebraically.
Trigonometry LESSON SEVEN - Trigonometric Identities II Example 4 Proving the sum and difference identities. a) Explain how to construct the diagram shown. Enrichment Example Students who plan on taking university calculus should complete this example. D C F β α A E G B b) Explain the next steps in the construction. D α C H F β α A E G B
Trigonometry LESSON SEVEN- Trigonometric Identities II c) State the side lengths of all the triangles. D D D α H F F α+β β F A E A A α G d) Prove the sum and difference identity for sine.
n P r = n! (n - r)! Permutations and Combinations LESSON ONE - Permutations Example Introduction to Permutations. Permutations Three letters (A, B, and C) are taken from a set of letter tiles and arranged to form words. In this question, ACB counts as a word - even though it s not an actual English word. a) Use a tree diagram to find the number of unique words. A B C b) Use the Fundamental Counting Principle to find the number of unique words. c) Use permutation notation to find the number of unique words. Evaluate using a calculator. d) What is meant by the terms single-case permutation and multi-case permutation? e) Use permutations to find the number of ways a one-, two-, or three-letter word can be formed.
Permutations and Combinations LESSON ONE - Permutations n P r = n! (n - r)! Example Evaluate each of the following factorial expressions. a) 4! b)! c) 0! d) (-)! Factorial Notation 5! e) f)! 8! 7!! g) n! (n - )! h) (n + )! (n - )!
n P r = n! (n - r)! Permutations and Combinations LESSON ONE - Permutations Example Permutations with Repetitions NOT Allowed. (Finite Sample Sets) a) A Grade student is taking Biology, English, Math, and Physics in her first term. If a student timetable has room for five courses (meaning the student has a spare), how many ways can she schedule her courses? i) Fundamental Counting Principle ii) Permutation Notation Simple Permutations Block Block Block Block Block 4 Block 5 Single-Case Permutations One Possible Timetable Course Math 0- Spare Physics 0 English 0- Biology 0 b) A singing competition has three rounds. In each round, the singer has to perform one song from a particular genre. How many different ways can the performer select the genres? i) Fundamental Counting Principle ii) Permutation Notation Round Round Round Rock Metal Punk Alternative Pop Dance Country Blues Folk c) A web development team of three members is to be formed from a selection pool of 0 people. The team members will be assigned roles of programmer, graphic designer, and database analyst. How many unique teams are possible? You can assume that each person in the selection pool is capable of performing each task. i) Fundamental Counting Principle ii) Permutation Notation d) There are letter tiles in a bag, and no letter is repeated. Using all of the letters from the bag, a six-letter word, a five-letter word, and a two-letter word are made. How many ways can this be done? i) Fundamental Counting Principle ii) Permutation Notation
Permutations and Combinations LESSON ONE - Permutations n P r = n! (n - r)! Example 4 Permutations with Repetitions NOT Allowed. (Finite Sample Sets) Repetitions NOT Allowed a) How many ways can the letters in the word SEE be arranged? i) Tree Diagram ii) Fundamental Counting Principle Single-Case Permutations b) How many ways can the letters in the word MISSISSAUGA be arranged? c) A multiple-choice test has 0 questions. Three questions have an answer of A, four questions have an answer of B, one question has an answer of C, and two questions have an answer of D. How many unique answer keys are possible? B d) How many pathways exist from point A to point B if the only directions allowed are north and east? A e) How many ways can three cars (red, green, blue) be parked in five parking stalls? f) An electrical panel has five switches. How many ways can the switches be positioned up or down if three switches must be up and two must be down? One possible switch arrangement.
n P r = n! (n - r)! Permutations and Combinations LESSON ONE - Permutations Example 5 Permutations where Repetitions ARE Allowed. (Infinite Sample Sets) Repetitions ARE Allowed a) There are three switches on an electrical panel. How many unique up/down sequences are there? Single-Case Permutations One possible switch arrangement. b) How many two-letter words can be created using the letters A, B, C, and D? c) A coat hanger has four knobs, and each knob can be painted any color. If six different colors of paint are available, how many ways can the knobs be painted? d) A phone number in British Columbia consists of one of four area codes (6, 50, 604, and 778), followed by a 7-digit number that cannot begin with a 0 or. How many unique phone numbers are there? e) An identification code consists of any two letters followed by any three digits. How many identification codes can be created?
Permutations and Combinations LESSON ONE - Permutations n P r = n! (n - r)! Example 6 Permutations with Repetitions NOT Allowed. (Finite Sample Sets) Constraints and Line Formations Six people (Andrew, Brenda, Cory, Danielle, Eliza, Frank) are going to be seated in a line. How many unique lines can be formed if: Single-Case Permutations a) Frank must be seated in the third chair? b) Brenda or Cory must be in the second chair, and Eliza must be in the third chair? c) Danielle can t be at either end of the line? d) men and women alternate positions, with a woman sitting in the first chair? e) the line starts with the pattern man-man-woman?
n P r = n! (n - r)! Permutations and Combinations LESSON ONE - Permutations Example 7 Permutations with Repetitions NOT Allowed. (Finite Sample Sets) Constraints and Words How many ways can you order the letters from the word TREES if: Single-Case Permutations a) a vowel must be at the beginning? b) it must start with a consonant and end with a vowel? c) the R must be in the middle? d) it begins with exactly one E? e) it ends with TR? f) consonants and vowels alternate?
Permutations and Combinations LESSON ONE - Permutations n P r = n! (n - r)! Example 8 Permutations with Repetitions NOT Allowed. (Finite Sample Sets) Objects ALWAYS Together a) How many ways can chemistry books, 4 math books, and 5 physics books be arranged if books on each subject must be kept together? Single-Case Permutations b) How many arrangements of the word ACTIVE are there if C&E must always be together? c) How many arrangements of the word ACTIVE are there if C&E must always be together, and in the order CE? d) Six people (Andrew, Brenda, Cory, Danielle, Eliza, Frank) are going to be seated in a line. How many unique lines can be formed if Cory, Danielle, and Frank must be seated together?
n P r = n! (n - r)! Permutations and Combinations LESSON ONE - Permutations Example 9 Permutations with Repetitions NOT Allowed. (Finite Sample Sets) a) How many ways can the letters in QUEST be arranged if the vowels must never be together? Objects NEVER Together i) Use a shortcut that works for separating two items. ii) Use a general method. Single-Case Permutations b) Eight cars ( red, blue, and yellow) are to be parked in a line. How many unique lines can be formed if the yellow cars must not be together? Assume that cars of each color are identical. i) Use a shortcut that works for separating two items. ii) Use a general method. c) How many ways can the letters in READING be arranged if the vowels must never be together?
Permutations and Combinations LESSON ONE - Permutations n P r = n! (n - r)! Example 0 More Than One Case. (At Least/At Most) Multi-Case Permutations a) How many words (with at most three letters) can be formed from the letter tiles SUNDAY? b) How many words (with at least five letters) can be formed from the letter tiles SUNDAY? c) How many -digit odd numbers greater than 600 can be formed using the digits,, 4, 5, 6, and 7, if a number contains no repeating digits? d) Six vehicles ( different brands of cars and different brands of trucks) are going to be parked in a line. How many unique lines can be formed if the row starts with at least two trucks? e) Six vehicles ( different brands of cars and different brands of trucks) are going to be parked in a line. How many unique lines can be formed if trucks and cars alternate positions?
n P r = n! (n - r)! Permutations and Combinations LESSON ONE - Permutations Example a) Evaluate 4 P Permutation Formula: Solve each of the following without using a calculator. P = n! n r (n - r)! b) Evaluate P 5! c) Write as a permutation.! d) Write! as a permutation.
Permutations and Combinations LESSON ONE - Permutations n P r = n! (n - r)! a) Example n! (n - )! = 5n Equations with Factorials and Permutations. Solve each of the following without using a calculator. b) (n + )! = n! n P r = n! (n - r)! n! c) 0 = P n- n- d) (n + )! (n - )! = 4n +
n P r = n! (n - r)! Permutations and Combinations LESSON ONE - Permutations Example a) n P = 56 Equations with Factorials and Permutations. Solve each of the following without using a calculator. b) 6 P r = 0 n P r = n! (n - r)! c) n + P = 0 d) n - P = n - 4 P
Permutations and Combinations LESSON ONE - Permutations n P r = n! (n - r)! This page has been left blank for correct workbook printing.
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example Introduction to Combinations. Combinations There are four marbles on a table, and each marble is a different color (red, green, blue, and yellow). Two marbles are selected from the table at random and put in a bag. a) Is the order of the marbles, or the order of their colors, important? b) Use a tree diagram to find the number of unique color combinations for the two marbles. c) Use combination notation to find the number of unique color combinations. d) What is meant by the terms single-case combination and multi-case combination? e) How many ways can three or four marbles be chosen?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! Example Combinations with Repetitions NOT Allowed. (Finite Sample Sets) Sample Sets with NO Subdivisions Single-Case Combinations a) There are five toppings available for a pizza (mushrooms, onions, pineapple, spinach, and tomatoes). If a pizza is ordered with three toppings, and no topping may be repeated, how many different pizzas can be created? b) A committee of 4 people is to be formed from a selection pool of 9 people. How many possible committees can be formed? c) How many 5-card hands can be made from a standard deck of 5 cards? 5 6 A 4 d) There are 9 dots randomly placed on a circle. i) How many lines can be formed within the circle by connecting two dots? ii) How many triangles can be formed within the circle?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example Combinations with Repetitions NOT Allowed. (Finite Sample Sets) Sample Sets with Subdivisions Single-Case Combinations a) How many 6-person committees can be formed from men and 9 women if men and women must be on the committee? b) A crate of toy cars contains 0 working cars and 4 defective cars. How many ways can 5 cars be selected if only work? c) From a deck of 5 cards, a 6-card hand is dealt. How many distinct hands are there if the hand must contain spades and diamonds? d) A bouquet contains four types of flowers: Flower Type Focal Flowers: Large and eye-catching flowers that draw attention to one area of the bouquet. Fragrant Flowers: Flowers that add a pleasant fragrance to the bouquet. Line Flowers: Tall and narrow flowers used to establish the height of a floral bouquet. Filler Flowers: Unobtrusive flowers that give depth to the bouquet. Examples Roses, Peonies, Hydrangeas, Chrysanthemums, Tulips, and Lilies Petunia, Daffodils, Daphnes, Gardenia, Lilacs, Violets, Magnolias Delphiniums, Snapdragons, Bells of Ireland, Gladioli, and Liatris Daisies, Baby's Breath, Wax Flowers, Solidago, and Caspia A florist is making a bouquet that uses one type of focal flower, no fragrant flowers, three types of line flowers and all of the filler flowers. How many different bouquets can be made?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! Example 4 Combinations with Repetitions NOT Allowed. (Finite Sample Sets) More Sample Sets with Subdivisions Single-Case Combinations a) A committee of 5 people is to be formed from a selection pool of people. If Carmen must be on the committee, how many unique committees can be formed? b) A committee of 6 people is to be formed from a selection pool of people. If Grant and Helen must be on the committee, but Aaron must not be on the committee, how many unique committees can be formed? c) Nine students are split into three equal-sized groups to work on a collaborative assignment. How many ways can this be done? Does the sample set need to be subdivided in this question? d) From a deck of 5 cards, a 5-card hand is dealt. How many distinct 5-card hands are there if the ace of spades and two of diamonds must be in the hand? e) A lottery ticket has 6 numbers from -49. Duplicate numbers are not allowed, and the order of the numbers does not matter. How many different lottery tickets contain the numbers, 4 and 48, but exclude the numbers 0 and 40?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example 5 Combinations with Repetitions NOT Allowed. (Finite Sample Sets) Permutations and Combinations Together Single-Case Combinations a) How many five-letter words using letters from TRIANGLE can be made if the five-letter word must have two vowels and three consonants? b) There are 4 men and 5 women on a committee selection pool. A three-person committee consisting of President, Vice-President, and Treasurer is being formed. How many ways can exactly two men be on the committee? c) A music teacher is organizing a concert for her students. If there are six piano students and seven violin students, how many different concert programs are possible if four piano students and three violin students perform in an alternating arrangement?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! Example 6 Combinations with Repetitions NOT Allowed. (Finite Sample Sets) Handshakes, Teams, and Shapes. Single-Case Combinations a) Twelve people at a party shake hands once with everyone else in the room. How many handshakes took place? b) If each of the 8 teams in a league must play each other three times, how many games will be played? (Note: This is a multi-case combination) c) If there are 8 dots on a circle, how many quadrilaterals can be formed? d) A polygon has 6 sides. How many diagonals can be formed?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example 7 Combinations where Repetitions ARE Allowed. (Infinite Sample Sets) Single-Case Combinations a) A jar contains quarters, loonies, and toonies. If four coins are selected from the jar, how many unique coin combinations are there? b) A bag contains marbles with four different colors (red, green, blue, and yellow). If three marbles are selected from the bag, how many unique color combinations are there?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! Example 8 More Than One Case (At Least/At Most). a) A committee of 5 people is to be formed from a group of 4 men and 5 women. How many committees can be formed if at least women are on the committee? Multi-Case Combinations b) From a deck of 5 cards, a 5-card hand is dealt. How many distinct hands can be formed if there are at most queens?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations c) From a deck of 5 cards, a 5-card hand is dealt. How many distinct hands can be formed if there is at least red card? d) A research team of 5 people is to be formed from biologists, 5 chemists, 4 engineers, and programmers. How many teams have exactly one chemist and at least engineers? e) In how many ways can you choose one or more of 5 different candies?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! Example 9 a) Evaluate 7 C 5 Combination Formula. n C r = n! (n - r)!r! b) Evaluate C c) Evaluate 6! d) Write as a combination. 4!! e) Write as a combination. 5! 4!
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example 0 Combination Formula. Solve for the unknown algebraically. a) n C = b) 4 C r = 6 n C r = n! (n - r)!r! c) d)
Permutations and Combinations LESSON TWO - Combinations Example n C 4 Combination Formula. Solve for the unknown algebraically. a) = b) n - C n C r n C n - r = n C r = n! (n - r)!r! n C r = n! (n - r)!r! c) P = C d) C = n - n - C n + n +
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example Assorted Mix I Assorted Mix I a) A six-character code has the pattern shown below, and the same letter or digit may be used more than once. How many unique codes can be created? Letter Digit Digit Digit Digit Letter STRATEGY: Organize your thoughts with these guiding questions: ) Permutation or Combination? ) Single-Case or Multi-Case? ) Are repetitions allowed? 4) What is the sample set? Are there subdivisions? 5) Are there any tricks or shortcuts? b) If there are different parkas, 5 different scarves, and 4 different tuques, how many winter outfits can be made if an outfit consists of one type of each garment? c) If a 5-card hand is dealt from a deck of 5 cards, how many hands have at most one diamond? d) If there are three cars and four motorcycles, how many ways can the vehicles park in a line such that cars and motorcycles alternate positions?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! e) Show that n C r = n C n - r. f) There are nine people participating in a raffle. Three $50 gift cards from the same store are to be given out as prizes. How many ways can the gift cards be awarded? g) There are nine competitors in an Olympic event. How many ways can the bronze, silver, and gold medals be awarded? h) A stir-fry dish comes with a base of rice and the choice of five toppings: broccoli, carrots, eggplant, mushrooms, and tofu. How many different stir-fry dishes can be prepared if the customer can choose zero or more toppings?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example Assorted Mix II Assorted Mix II a) A set of tiles contains eight letters, A - H. If two of these sets are combined, how many ways can all the tiles be arranged? Leave your answer as an exact value. b) A pattern has five dots such that no three points are collinear. How many lines can be drawn if each dot is connected to every other dot? c) How many ways can the letters in CALGARY be arranged if L and G must be separated? d) A five-person committee is to be formed from people. If Ron and Sara must be included, but Tracy must be excluded due to a conflict of interest, how many committees can be formed?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! e) Moving only south and east, how many unique pathways connect points A and C? A B C f) How many ways can the letters in SASKATOON be arranged if the letters K and T must be kept together, and in that order? g) A 5-card hand is dealt from a deck of 5 cards. How many hands are possible containing at least three hearts? h) A healthy snack contains an assortment of four vegetables. How many ways can one or more of the vegetables be selected for eating?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example 4 Assorted Mix III Assorted Mix III a) How many ways can the letters in EDMONTON be arranged if repetitions are not allowed? b) A bookshelf has n fiction books and six non-fiction books. If there are 50 ways to choose two books of each type, how many fiction books are on the bookshelf? c) How many different pathways exist between points A and D? B A D d) How many numbers less than 60 can be made using only the digits, 5, and 8, if the numbers formed may contain repeated digits? C
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! e) A particular college in Alberta has a list of approved pre-requisite courses: Math Science English Other Math 0- or Math 0- Biology 0 Chemistry 0 Physics 0 English 0- Option A Option B Option C Option D Option E Five courses are required for admission to the college. Math 0- (or Math 0-) and English 0- are mandatory requirements, and at least one science course must be selected as well. How many different ways could a student select five courses on their college application form? f) How many ways can four bottles of different spices be arranged on a spice rack with holes for six spice bottles? g) If there are 8 rock songs and 9 pop songs available, how many unique playlists containing rock songs and pop songs are possible? h) A hockey team roster contains forwards, 6 defencemen, and goalies. During play, only six players are allowed on the ice - forwards, defencemen, and goalie. How many different ways can the active players be selected?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example 5 Assorted Mix IV Assorted Mix IV a) A fruit mix contains blueberries, grapes, mango slices, pineapple slices, and strawberries. If six pieces of fruit are selected from the fruit mix and put on a plate, how many ways can this be done? b) How many ways can six letter blocks be arranged in a pyramid, if all of the blocks are used? A B C D E F c) If a 5-card hand is dealt from a deck of 5 cards, how many hands have cards that are all the same color? d) If a 5-card hand is dealt from a deck of 5 cards, how many hands have cards that are all the same suit?
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! e) A multiple choice test contains 5 questions, and each question has four possible responses. How many different answer keys are possible? f) How many diagonals are there in a pentagon? g) How many ways can eight books, each covering a different subject, be arranged on a shelf such that books on biology, history, or programming are never together? h) If a 5-card hand is dealt from a deck of 5 cards, how many hands have two pairs?
n C r = n! (n - r)!r! Permutations and Combinations LESSON TWO - Combinations Example 6 Assorted Mix V Assorted Mix V a) How many ways can six people be split into two equal-sized groups? b) Show that 5! + 6! = 7 5! c) Five different types of fruit and six different types of vegetables are available for a healthy snack tray. The snack tray is to contain two fruits and three vegetables. How many different snack trays can be made if blueberries or carrots must be served, but not both together? d) In genetics, a codon is a sequence of three letters that specifies a particular amino acid. A fragment of a particular protein yields the amino acid sequence: Met - Gly - Ser - Arg - Cys - Gly. How many unique codon arrangements could yield this amino acid sequence? Amino Acid Arginine (Arg) Cysteine (Cys) Glycine (Gly) Methionine (Met) Serine (Ser) Codon(s) CGU, CGC, CGA, CGG, AGA, AGG UGU, UGC GGU, GGC, GGA, GGG AUG UCU, UCC, UCA, UCG, AGU, AGC
Permutations and Combinations LESSON TWO - Combinations n C r = n! (n - r)!r! e) In a tournament, each player plays every other player twice. If there are 56 games, how many people are in the tournament? f) The discount shelf in a bookstore has a variety of books on computers, history, music, and travel. The bookstore is running a promotion where any five books from the discount shelf can be purchased for $0. How many ways can five books be purchased? g) Show that n C r + n C r + = n + C r +. Note: This question will require more paper than is provided on this page. h) How many pathways are there from point A to point C, passing through point B? Each step of the pathway must be getting closer to point C. C B A
t k+ = n C k (x) n-k (y) k Permutations and Combinations LESSON THREE - The Binomial Theorem Example Pascal s Triangle Pascal s Triangle Pascal s Triangle is a number pattern with useful applications in mathematics. Each row is formed by adding together adjacent numbers from the preceding row. a) Determine the eighth row of Pascal s Triangle. First seven rows of Pascal s Triangle. 4 6 4 5 0 0 5 6 5 0 5 6 b) Rewrite the first seven rows of Pascal s Triangle, but use combination notation instead of numbers. c) Using the triangles from parts (a & b) as a reference, explain what is meant by n C k = n C n - k.
Permutations and Combinations LESSON THREE - The Binomial Theorem t k+ = n C k (x) n-k (y) k Example Rows and Terms of Pascal s Triangle. Pascal s Triangle a) Given the following rows from Pascal s Triangle, write the circled number as a combination. i) 8 8 56 70 56 8 8 ii) 66 0 495 79 94 79 495 0 66 b) Use a combination to find the third term in row of Pascal s Triangle. c) Which positions in the th row of Pascal s Triangle have a value of 65? d) Find the sum of the numbers in each of the first four rows of Pascal s Triangle. Use your result to derive a function, S(n), for the sum of all numbers in the n th row of Pascal s Triangle. What is the sum of all numbers in the eleventh row?
t k+ = n C k (x) n-k (y) k Permutations and Combinations LESSON THREE - The Binomial Theorem Example Use Pascal s Triangle to determine the number of paths from point A to point B if east and south are the only possible directions. Pascal s Triangle and Pathways a) b) A A B B c) d) A A B B
Permutations and Combinations LESSON THREE - The Binomial Theorem t k+ = n C k (x) n-k (y) k Example 4 The Binomial Theorem. The Binomial Theorem a) Define the binomial theorem and explain how it is used to expand (x + ). Expand the expressions in parts (b) and (c) using the binomial theorem. b) (x + ) 6 c) (x - ) 4
t k+ = n C k (x) n-k (y) k Permutations and Combinations LESSON THREE - The Binomial Theorem Example 5 Expand each expression. The Binomial Theorem a) (x - y) 4 b) c)
Permutations and Combinations LESSON THREE - The Binomial Theorem t k+ = n C k (x) n-k (y) k Example 6 Write each expression as a binomial power. Condense to a Binomial a) x 4 + 4x y + 6x y + 4xy + y 4 b) a 5-40a 4 b + 70a b - 080a b + 80ab 4-4b 5 c)
t k+ = n C k (x) n-k (y) k Permutations and Combinations LESSON THREE - The Binomial Theorem Example 7 Use the general term formula to find the requested term in a binomial expansion. a) Find the third term in the expansion of (x - ) 4. General Term t k + = n C k (x) n - k (y) k b) Find the fifth term in the expansion of (a - b ) 8. c) Find the fourth term in the expansion of x -. x 6
Permutations and Combinations LESSON THREE - The Binomial Theorem t k+ = n C k (x) n-k (y) k Example 8 Answer each of the following questions. Finding Specific Values a) In the expansion of (5a - b) 9, what is the coefficient of the term containing a 5? b) In the expansion of (4a + b ) 5, what is the coefficient of the term containing b? c) In the expansion of (a - 4) 8, what is the middle term? d) If there are terms are in the expansion of (a - ) k-5, what is the value of k?
t k+ = n C k (x) n-k (y) k Permutations and Combinations LESSON THREE - The Binomial Theorem Example 9 Answer each of the following questions. Finding Specific Values a) A term in the expansion of (ma - 4) 5 is -5760a. What is the value of m? b) The term -080a b occurs in the expansion of (a - b) n. What is the value of n? a c) A term in the expansion of (a + m) 7 is. What is the value of m. b
Permutations and Combinations LESSON THREE - The Binomial Theorem t k+ = n C k (x) n-k (y) k Example 0 Answer each of the following questions. Finding Specific Values a) In the expansion of, what is the constant term? b) In the expansion of, what is the constant term? c) In the expansion of b, one of the terms is 40x. What is the value of b?
Example : Trigonometry Lesson One: Degrees and Radians a) The rotation angle between the initial arm and the terminal arm is called the standard position angle. θ b) An angle is positive if we rotate the terminal arm counterclockwise, and negative if rotated clockwise. 0 c) The angle formed between the terminal arm and the x-axis is called the reference angle. 50 d) If the terminal arm is rotated by a multiple of 60 in either direction, it will return to its original position. These angles are called co-terminal angles. e) A principal angle is an angle that exists between 0 and 60. 40 60 Answer Key Note: For illustrative purposes, all diagram angles will be in degrees. f) The general form of co-terminal angles is θ c = θ p + n(60 ) using degrees, or θ c = θ p + n() using radians. 45, 405, 765, 5, 485 45, -5, -675, -05, -95-0 0 Example : a) i. One degree is defined as /60 th of a full rotation. ii. One radian is the angle formed when the terminal arm swipes out an arc that has the same length as the terminal arm. One radian is approximately 57.. 57. iii. One revolution is defined as 60º, or pi. It is one complete rotation around a circle. Conversion Multiplier Reference Chart degree radian revolution rev degree 80 60 radian revolution 80 60 rev rev rev b) i. 0.40 rad ii. 0.06 rev iii. 48.97 iv. 0.4 rev v. 70 vi. 4.7 rad c) i. 0.79 rad ii. /4 rad Example : a).05 rad b) 7/6 rad c) / rev d) 4.4 e) 70 f) 4.7 rad g) /4 rev h) 80 i) 6 rad Example 4: = 0 = 5 = 50 90 = 60 = 45 = 0 = Example 5: a) θ r = 0 b) θ r = 80 c) θ r = 56 (or 0.98 rad) -60 d) θ r = 45 (or /4 rad) 45 5 e) θ r = 5 (or /7 rad) = 80 0 = 60 = 0 0 θ 80 θ θ 56 04 θ θ 09 5 = 0 = 5 = 40 00 = 5 = 0 = Example 6: a) θ p = 0, θ r = 0 b) θ p = 5, θ r = 45 c) θ p = 56, θ r = 4 d) θ p = 0, θ r = 60 (or θ p =.7, θ r = 0.4) (or θ p = /, θ r = /) Example 7: = 70 0 0 a) θ = 60, θ = 60 p b) θ = -495, θ = 5 p θ c = -00, 40, 780 θ c = -855, -5, 5, 585 60 θ 0 56 60 4 θ θ θ 45 5 Example 8: a) θ p = 9 b) θ p = 48 c) θ p = 44 d) θ p = 0 (or.58 rad) (or 4/5 rad) (or /6 rad) 9 5 48 44 c) θ = 675, θ = 5 p d) θ = 480, θ = 0 p θ c = -45, 5 θ c = -960, -600, -40, 0, (or θ c = -0.785, 5.50) 840, 00 5 0 (or θ c = -6/, -0/, -4/, /, 4/, 0/) Example 9: a) θ c = 80 b) θ c = -8/5 c) θ c = 0 d) θ c = / 0
Answer Key Example 0: a) θ p =.6, θ r = 67.8.6 b) θ p = 0.69, θ r = 56. θ -5 67.8 56. θ - 0.69 Example : a) sinθ: QI: +, QII: +, QIII: -, QIV: - b) cosθ: QI +, QII: -, QIII: -, QIV: + c) tanθ: QI +, QII: -, QIII: +, QIV: - d) cscθ: QI: +, QII: +, QIII: -, QIV: - e) secθ: QI +, QII: -, QIII: -, QIV: + f) cotθ: QI +, QII: -, QIII: +, QIV: - g) sinθ & cscθ share the same quadrant signs. cosθ & secθ share the same quadrant signs. tanθ & cotθ share the same quadrant signs Example : a) i. QIII or QIV ii. QI or QIV iii. QI or QIII b) i. QI ii. QIV iii. QIII c) i. none ii. QIII iii. QI Example : a) θ p = 0.6, θ r =.6 b) θ p = 54.6, θ r = 5.8 (or θ p =.54 rad, θ r = 0.9 rad) (or θ p =.70 rad, θ r = 0.44 rad) 54.6-5 0.6 - θ.6 7 θ 5.8 Example 4: a) θ p =., θ r = 6.87 b) θ p = 6., θ r =.69 4 6.87 θ 5 -.69 θ -. 6. Example 5: a) If the angle θ could exist in either quadrant or... I or II I or III I or IV II or III II or IV III or IV The calculator always picks quadrant I I I II IV IV b) Each answer is different because the calculator is unaware of which quadrant the triangle is in. The calculator assumes Mark s triangle is in QI, Jordan s triangle is in QII, and Dylan s triangle is in QIV. Example 6: a) The arc length can be found by multiplying the circumference by the sector percentage. This gives us: a = r θ/ = rθ. b).5 cm c) 4.59 d).46 cm e) n = 7/6 Example 7: a) The area of a sector can be found by multiplying the area of the full circle by the sector percentage to get the area of the sector. This gives us: a = r θ/ = r θ/. b) 8/ cm c) cm d) 8/ cm e) 5 cm Example 8: a) 600 /s b) 0.07 rad/s c).04 km d) 70 rev/s e).60 rev/s Example 9: a) /700 rad/s b) 468.45 km
Answer Key Trigonometry Lesson Two: The Unit Circle Example : a) i. ii. b) i. Yes ii. No 0 0 (0.6, 0.8) (0.5, 0.5) c) i. ii. -0 0-0 0 iii. y = 0 iv. Example : See Video. Example : a) b) - c) d) e) 0 f) 0 g) h) Example 5: a) b) Example 6: a) b),,,, Example 7: See Video. Example 4: a) b) c) d) e) - f) g) h),,,,,,,,,,,, Example 4: a) C = b) The central angle and arc length of the unit circle are equal to each other. c) a = / d) a = 7/6 Example 5: a) The unit circle and the line y = do not intersect, so it's impossible for sinθ to equal. b) Range Number Line cosθ & sinθ cscθ & secθ tanθ & cotθ c) d) 5., 0.70 e) - - - 0 0 0 y = Example 8: a) - b) undefined c) d) e) f) - g) 0 h) Example 9: a) b) c) d) Example 0: a) b) c) d) Example : a) - b) c) undefined d) undefined Example 6: a) Inscribe a right triangle with side lengths of x, y, and a hypotenuse of into the unit circle. We use absolute values because technically, a triangle must have positive side lengths. Plug these side lengths into the Pythagorean Theorem to get x + y =. b) Use basic trigonometric ratios (SOHCAHTOA) to show that x = cosθ and y = sinθ. c) θ p = 67., θ r =.68 x y Example : See Video. Example : a) P(/) means "point coordinates at /". b) c) d) e) P() = (-0.9900, 0.4) Example 7: a) (67, ) b) (-79, ) Example 8: a) See Video b) 60 m
Answer Key Trigonometry Lesson Three: Trigonometric Functions I Example : a) (-5/6, ), (-/6, -4), (7/6, ) b) (-/4, -), (/4, 6), (7/4, -8) c) (-6, 8), (-, -8), (4, -4) d) (-, 0), (/, -0), (5/, -0) Example : a) y = sinθ b) a = c) P = d) c = 0 e) d = 0 f) θ = n, nεi g) (0, 0) h) Domain: θ ε R, Range: - y y Example : a) y = cosθ b) a = c) P = d) c = 0 e) d = 0 f) θ = / + n, nεi g) (0, ) h) Domain: θ ε R, Range: - y y 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0 7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0 7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ - - Example 4: a) y = tanθ b) Tangent graphs do not have an amplitude. c) P = d) c = 0 e) d = 0 f) θ = n, nεi g) (0, 0) h) Domain: θ ε R, θ / + n, nεi, Range: y ε R y Example 5: a) b) 5 5 7 5 4 5 7 5 5 7 5 4 5 7 6 4 4 6 6 4 4 6 6 4 4 6 6 4 4 6 θ - 0 0 - -5-5 Example 7: c) d) a) b) c) d) 5 5 5 5 5 5 0 0 0 0 0 0-5 -5-5 -5-5 -5 Example 6: Example 8: a) b) c) - d) 5 a) b) Example 9: a) b) c) d) c) d) 0 0 0 4 6 0 4 6 8 0 - - - -
Answer Key Example 0: Example : a) b) a) b) 6 - - - - - 4 6-6 - - - - c) d) c) d) 4 - - 0 4 6 - - - 4 - - - - - Example : a) b) Example 4: c) d) a) b) c) d) Example : a) b) Example 5: a) b) 4-4 - 4-4 - 5 - - c) d) 0 4 6 0 4-5 - c) d) - - - - 5 6 - -4 0 0-5 -6 Example 6: a) b)
Answer Key Example 7: a) y = secθ b) P = c) Domain: θ ε R, θ / + n, nεi; Range: y -, y d) θ = / + n, nεi y y 0 θ 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0-7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ - - - Example 8: a) y = cscθ b) P = c) Domain: θ ε R, θ n, nεi; Range: y -, y d) θ = n, nεi y y 0 θ 5 7 5 4 5 7 6 4 4 6 6 4 4 6 0-7 5 4 5 7 5 6 4 4 6 6 4 4 6 θ - - - Example 9: a) y = cotθ b) P = c) Domain: θ ε R, θ n, nεi; Range: yεr d) θ = n, nεi y y 0 θ 7 5 4 5 7 5 5 7 5 4 5 7 6 4 4 6 6 4 4 6 6 4 4 6 6 4 4 6 θ - - - Example 0: a) b) c) d) 0 0 0 0 - - - - Domain: θ ε R, θ / + n, nεi; (or: θ ε R, θ / ± n, nεw) Range: y -/, y / Domain: θ ε R, θ /4 + n/, nεi; (or: θ ε R, θ /4 ± n/, nεw) Range: y -, y Domain: θ ε R, θ /4 + n, nεi; (or: θ ε R, θ /4 ± n, nεw) Range: y -, y Domain: θ ε R, θ n(), nεi; (or: θ ε R, θ ±n(), nεw) Range: y ε R
Answer Key Trigonometry Lesson Four: Trigonometric Functions II Example : a) b) Example : a) b) y y h y θ 80º 60º 540º θ 5 0 45 60 t 8 6 4 x - - - - - - - - Example : Example 5: a) b) a) - -4-6 -8-0 - -4-6 -8-0 - -4-6 y 4 5 6 7 8 x -5-75 -5-75 -5-00 -50-00 -50 y 75 50 5 0 5 75 5 75 5 75 50 00 50 00 50 x b) c) d) Example 6: a) Example 4: a) b) b) c) d) e) 5 y 0 8 6 4 0 8 6 4 0 - -4-6 -8 0 0 0 40 50 60 70 80 90 00 0 x 0 - - - y 0 8 6 4 0 8 6 4 4 5 x Example 7: a) b) c) d) Example 8: a) b) The b-parameter is doubled when the period is halved. The a, c, and d parameters remain the same. c) The d-parameter decreases by units, giving us d = 4. All other parameters remain unchanged.
Mice Answer Key Example 9: Example : a) h(t) 05 75 45 a) Decimal daylight hours: 6.77 h,.8 h, 7.8 h,.8 h, 6.77 h b) d(n) 0 6 = 5.55cos + +.95 65 c) d ( n) ( n ) 4 d) 5.86 h e) 64 days 5 0 t 8 b) c) If the wind turbine rotates counterclockwise, we still get the same graph. Example 0: a) h(θ) 4-50 0 50 00 50 00 50 00 50 400 n Example 4: a) Decimal hours past midnight:.0 h, 8.0 h, 4.0 h, 0.0 h b) h(t) 6 c) d) 0.75 m e).% 8 4 90 θ b), Example : a) h(t) 5. 4.0 c) The angle of elevation increases quickly at first, but slows down as the helicopter reaches greater heights. The angle never actually reaches 90. Example 5: 0 4 8 6 0 4 t a) Population b) See Video. 6000 M(t) 000 Owls 8000 00 O(t) 50 00.8 0 4 5 6 7 8 Time (years) b) c).86 m t d) 0.6 s Example 6:.5 m h(t).0 (8,.5).5 Example : a) h(t) 5 0 45 60 t 6 Example 7: 5.6 s and 8. s h(t) 9 t 0 5 50 75 00 5 50 75 00 0 (5.6, 0.5) (8.,.) b) c) 8.4 m d) 6.78 s 5 0 45 60 t
Answer Key Trigonometry Lesson Five: Trigonometric Equations Note: n ε I for all general solutions. Example : a), b), c) d) Example : a) b) c) - - - d) no solution e) f) - - - - - - - - Example : a) b) c) 90 90 90 50 0 50 45 80 0 0 0 60 80 0 0 0 60 80 45 45 0 60 0 5 70 70 70 Example 4: a) b) Intersection point(s) of original equation θ-intercepts Intersection point(s) of original equation θ-intercepts - - - - - - - - - - - -
Answer Key Example 5: a) b) c) d) 0 60 60 - - 40 - - - - Example 6: a) 97.46 and 4.54 b) 97.46 and 4.54 c) 97.46 and 4.54 d) 97.46 and 4.54 The unit circle is not useful for this question. 97.46 7.46 7.46 4.54 - - - - Example 7: - - a) b) c) Example 8: a) No Solution b) c) - - - - - - d) e) f) - - - - - -
Answer Key Example 9: a) b) c) 90 90 90 0 0 60 60 80 60 60 0 60 80 60 60 0 60 80 60 60 0 60 40 40 70 Example 0: 70 70 a) No Solution b) Intersection point(s) of original equation θ-intercepts Intersection point(s) of original equation θ-intercepts - - - - - - - - - - - - Example : a) b) c) d) 0 0 0 0 - - - - - - Example : a) b) c) d) 5 45 65 65 The unit circle is not useful for this question. - - - - - Example : - a) b) c) d)
Answer Key Example 4: a) b) c) d) - - - - Example 5: a) b) c) d) - - - - - - Example 6: a) b) c) - - - - - - - - - Example 7: Example 8: a) b) a) b) 4 4 6 8 - - - - Example 9: a) Example 0: a) Example : See Video b) Approximately days. Visible % b) See graph. c) 0.466 rad (or 6.6 ) d.00 8 0.50 4-4 0.466 θ 0 7 4 8 t -8
Answer Key Trigonometry Lesson Six: Trigonometric Identities I Note: n ε I for all general solutions. Example : a) Identity Equation - - - - - - b) i) ii) iii) iv) v) Not an Identity Not an Identity Identity Identity Not an Identity - - - - - - - - - - - - - Example : a) Use basic trigonometry (SOHCAHTOA) to show that x = cosθ and y = sinθ. θ x y b) Verify that the L.S. = R.S. for each angle. c) The graphs of y = sin x + cos x and y = are the same. - d) Divide both sides of sin x + cos x = by sin x to get + cot x = csc x. Divide both sides of sin x + cos x = by cos x to get tan x + = sec x. Example : a) b) e) Verify that the L.S. = R.S. for each angle. f) The graphs of y = + cot x and y = csc x are the same. - - - Example 5: a), b) The graphs of y = tan x + and y = sec x are the same. - - - - - - - Example 4: a) b) - - c) d) - - - - - - - -
Answer Key Example 6: a) b) - - c) d) 0 - Example 7: a) b) - - - - - - c) d) - - - - - - Example 8: a) b) - - - - - - c) d) - - - - - -
Answer Key Example 9: See Video Example 0: See Video Example : See Video Example : a) See Video b) c) d) Example : a) See Video b) c) d) Example 4: a) See Video b) c) d) The graphs are NOT identical. The R.S. has holes. The graphs are identical. The graphs are identical. - - - - - - - Example 5: a), b), - - - - - - c), d), 6 4 - -4-6 - Example 6: a), b), - 0 Note: All terms from the original equation were collected on the left side before graphing. - - -0 c), d), - -
Answer Key Example 7: a), b), 0 Note: All terms from the original equation were collected on the left side before graphing. 0 - -6 Note: All terms from the original equation were collected on the left side before graphing. -9-0 - c), d), - - - - Example 8: a) b) c) 7 4 - - 7 Example 9: See Video
Answer Key Trigonometry Lesson Seven: Trigonometric Identities II Note: n ε I for all general solutions. Example : a) b) c) d) e) f) Example : Example : a) b) c) a) b) c) d) See Video Example 4: Example 5: See Video a) b) c) Example 6: a) i. ii. 0 iii. undefined b) (answers may vary) i. c) (answers may vary) i. ii. iii. iv. ii. iii. iv. Example 0: a) b) d. 90 80 70 60 θ -. c) θ = 4.6 and θ = 65.4 Example : At 0, the cannonball hits the ground as soon as it leaves the cannon, so the horizontal distance is 0 m. At 45, the cannonball hits the ground at the maximum horizontal distance,. m. At 90, the cannonball goes straight up and down, landing on the cannon at a horizontal distance of 0 m Examples 7 - : Proofs. See Video. a) Example 4: a) b) Example 5: a) b) b) A 4900 The maximum area occurs when θ = 45. At this angle, the rectangle is the top half of a square. c) c) 45 90 θ d) d) c) i. ii. iii. Example : Example 6: a) Example 7: a) a) i. y = f(θ) + g(θ) b) i. 6 6 y = f(θ) + g(θ) b) b) 0 0 c) c) d) Example 8: 57 Example 9: 9.9 d) -6 ii. The waves experience constructive interference. iii. The new sound will be louder than either original sound. c) All of the terms subtract out leaving y = 0, A flat line indicating no wave activity. -6 ii. The waves experience destructive interference. iii. The new sound will be quieter than either original sound. Example : See Video. Example 4: See Video.
Answer Key Permutations and Combinations Lesson One: Permutations Example : a) Six words can be formed. b) = 6 c) P d) See Video e) P + P + P Example : a) 4 b) c) d) (-)! Does not exist. e) 0 f) 4 g) n n h) n + n Example : a) 0 b) 4 c) 70 d)! Example 4: a) b) 45800 c) 600 d) 0 e) 60 f) 0 Example 5: a) 8 b) 6 c) 96 d) 0 6 e) 676 000 Example 6: a) 0 b) 48 c) 480 d) 08 Example 7: a) 4 b) 8 c) d) 8 e) f) 6 Example 8: a) 0 680 b) 40 c) 0 d) 44 Example 9: a) 7 b) 40 c) 440 Example 0: a) 56 b) 440 c) 0 d) 44 e) 7 Example : a) 4 b) 0 c) 5 P d) P or P Example : a) n = 6 b) n = c) n = 5 d) n = Example : a) n = 8 b) r = c) n = d) n = 5 Permutations and Combinations Lesson Two: Combinations Example :a) The order of the colors is not important. b) 6 c) 4 C d) See Video e) 4 C + 4 C 4 Example : a) 0 b) 6 c) 598960 d) 6; 84 Example : a) 860 b) 70 c) 580008 d) 60 Example 4: a) 0 b) 70 c) 680 d) 9600 e) 44 Example 5: a) 600 b) 80 c) 75600 Example 6: a) 66 b) 84 c) 70 d) 9 Example 7: a) 5 b) 0 Example 8: a) 8 b) 594400 c) 580 d) 405 e) Example 9: a) b) c) 6 d) 6 C e) 5 C Example 0: a) n = 7 b) 4 C c) n = 5 d) n = 6 Example : a) n = 4 b) All n-values c) n = 4 d) n = 4 Example : a) 6760000 b) 40 c) 64500 d) 44 e) See Video f) 84 g) 504 h) Example : a) 6!/(!) 8 b) 0 c) 800 d) 56 e) 0 f) 5040 g) 4098 h) 5 Example 4: a) 0080 b) 5 c) 8 d) 9 e) 9 f) 60 g) 490 h) 6600 Example 5: a) 0 b) 70 c) 548 d) 560 e) 04 f) 5 g) 4400 h) 55 Example 6: a) 0 b) See Video c) 00 d) 5 e) n = 8 f) 56 g) See Video h) 6 Permutations and Combinations Lesson Three: The Binomial Theorem Example : a) The eighth row of Pascal's Triangle is:, 7,, 5, 5,, 7,. b) See Video. Note that rows and term positions use a zero-based index. c) There is symmetry in each row. For example, the second position of the sixth row is equal to the second-last position of the same row. Example : a) 8 C 0 ; C 0 b) C = 0 c) k = and 8, so the fourth and ninth positions have a value of 65. d) 04 Example : a) 0 b) 0 c) 66 d) 54 Example 4: a) The binomial theorem states that a binomial power of the form (x + y) n can be expanded into a series of terms with the form n C k x n-k y k, where n is the exponent of the binomial (and also the zero-based row of Pascal's Triangle), and k is the zero-based term position. b) c) Example 5: a) b) c) Example 6: a) b) c) Example 7: a) b) c) Example 8: a) b) c) d) Example 9: a) b) c) Example 0: a) b) c)