Mathematics Book Two. Trigonometry One Trigonometry Two Permutations and Combinations

Transcription

1 Mathematics 0- Book Two Trigonometry One Trigonometry Two Permutations and Combinations

2

3 A workbook and animated series by Barry Mabillard Copyright 04

4 This page has been left blank for correct workbook printing.

5 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

6 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)

7 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.

8 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.

9 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

10 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

11 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.

12 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 = = = = 0 = 5 = 40 0 = 5 = 00 = = 70

13 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) e) 7

14 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

15 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

16 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) Principal Angle of a Large Angle 9 c) d)

17 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.

18 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.

19 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 θ θ - θ θ - θ θ - θ θ - θ θ - θ θ d) cscθ e) secθ f) cotθ θ θ - θ θ - θ θ - θ θ - θ θ - θ θ 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)?

20 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 =

21 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)

22 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)

23 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

24 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

25 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 cm cm

26 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.

27 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 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.

28 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.

29 (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 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) - -

30 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.

31 (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.

32 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

33 (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

34 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 =

35 (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 =

36 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

37 (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

38 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)

39 (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)

40 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)

41 (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)

42 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.

43 (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

44 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

45 (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?

46 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.

47 (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.

48 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?

49 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

50 Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d c) y θ - d) y θ -40

51 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 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.

52 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 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.

53 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 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 = 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 = tan = tan = - = undefined

54 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 c) y = sinθ 5 d) y = cosθ

55 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 c) write a cosine function. d) write a cosine function ( ),

56 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 The d Parameter c) y = - sinθ + d) y = cosθ

57 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 c) write a cosine function. d) write a sine function

58 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 c) y = cos θ (0 θ 6) d) y = sin θ (0 θ 0)

59 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 c) y = cos θ - (- θ ) 4 d) y = sin θ (0 θ 6)

60 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

61 y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I c) write a sine function. The b Parameter d) write a sine function. 0 -

62 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) c) (- θ ) d) (- θ )

63 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) c) (- θ 4) d) (- θ )

64 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

65 y = asinb(θ - c) + d Trigonometry LESSON THREE - Trigonometric Functions I c) write a sine function. The c Parameter - - d) write a cosine function

66 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 ) c) - (0 θ ) d) (0 θ )

67 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) b) - - -

68 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 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 = sec = 6 - sec = undefined sec = sec = 4 sec = 6 sec = - sec 0 = - 0 θ 7 sec = sec = sec = - sec sec = 6 7 sec = 4 5 sec = = undefined

69 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 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 = csc = csc = - csc = - csc = csc = csc = -

70 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 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 = 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 = cot = cot = -

71 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) y = secθ - y = secθ c) d) y = cscθ - y = cotθ

72 Trigonometry LESSON THREE - Trigonometric Functions I y = asinb(θ - c) + d This page has been left blank for correct workbook printing.

73 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.

74 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 t 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?

75 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)

76 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)

77 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 b) write a sine function

78 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) (00, -50) -00

79 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.

80 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.

81 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.

82 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)

83 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)?

84 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?

85 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.

86 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.

87 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.

88 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

89 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.

90 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

91 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.

92 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 Owls Time (years) b) Describe how the graph shows the relationship between owl and mouse populations.

93 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? m

94 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) = t + 0t + If Tim throws the baseball 5 seconds after the ride begins, when are Shane and the ball at the same height?

95 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 =

96 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

97 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) b) c)

98 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

99 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

100 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

101 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)

102 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

103 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) b) c)

104 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

105 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

106 Trigonometry LESSON FIVE - Trigonometric Equations Example a) non-graphically, using the cos - feature of a calculator. Solve secθ = 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

107 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 +

108 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. - -

109 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

110 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

111 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: -

112 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:

113 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 % t b) In one cycle, for how many days is 60% or more of the moon s surface visible?

114 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)?

115 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 x

116 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 x x 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:

117 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)

118 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 = -

119 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

120 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) -

121 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) - - -

122 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) -

123 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) -

124 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) -

125 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) -

126 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) - - -

127 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) - - -

128 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) - - -

129 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) - - -

130 Trigonometry LESSON SIX- Trigonometric Identities I Example 9 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

131 Trigonometry LESSON SIX - Trigonometric Identities I Example 0 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

132 Trigonometry LESSON SIX- Trigonometric Identities I Example Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

133 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 -

134 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 = - - -

135 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 = - - -