4.1 Radian and Degree Measure


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1 Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position (of an angle): initial side is on the positive ais; positive angles rotate counterclockwise negative angles rotate clockwise Radian: a measure of length; 1 radian when r s r = radius, s = length of arc s r Arc Length: s r ( must be measured in radians) θ r s Recall: Circumference of a Circle; C r So, there are radians around the circle radians = 360, or radians = 180 Draw in the radians: Degrees and Radians: Coterminal Angles: angles with the same initial and terminal sides, for eample 360 or E1: Find a positive and negative coterminal angle for each. a. 130 b. 9 Page 1 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
2 Complementary Angles: The sum of two angles is or 90 Supplementary Angles: The sum of two angles is or 180 E: If possible find the complement and the supplement of a.) 5 b.) 4 5 Converting Degrees (D) to Radians (R): E3: Convert 540 to radians. D R (Note: 1 ) Note: We multiply by 180 so that the degrees cancel. This will help you remember what to multiply by. Converting from Radians (R) to Degrees (D): 180 D R d Or use the proportion: 180 r E4: Convert 3 radians to degrees. 4 Note: We multiply by 180 so that the s cancel. Page of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
3 Reference Angle: every angle has a reference angle, with initial side as the ais E.5 Draw and Find the reference angle of a.) 45 degrees, b.) 10 degrees, c.) 5 degrees, d.) 330 degrees. Special Angles to Memorize (Teacher Note: Have students fill this in for practice.) Degrees Radians Shortcut for Specials: Degrees to radians 1. Find the reference angle. Divide reference angle into angle 3. Write reference angle in radians times answer in # times Radians to degrees: 1. Write reference angle. Multiply eample 4 4 times Reflection: Date: 4.1 Radian and Degree Measure Continued Review: Convert 1. hours into hours and minutes. Solution: Convert 3 hours and 0 minutes into hours. Solution: Babylonian Number System: based on the number 60; 360 approimates the number of days in a year. Circles were divided into 360 degrees. A degree can further be divided into 60 minutes (60'), and each minute can be divided into 60 seconds (60''). Page 3 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
4 Converting from Degrees to DMS (Degrees Minutes Seconds): multiply by 60 E1: Convert to DMS. Converting to Degrees (decimal): divide by 60 E: Convert 7 13' 5'' to decimal degrees. Note: The degree and minute symbols can be found in the ANGLE menu. The seconds symbol can be found above the + sign (alpha +). Arc Length: s r ( must be measured in radians) E3: A circle has an 8 inch diameter. Find the length of an arc intercepted by a 40 central angle. Step One: Convert to radians. Step Two: Find the radius. Step Three: Solve for s. Linear Speed (eample: miles per hour): Angular Speed (eample: rotations per minute): s r l t t A t angle time length time E4: Find the linear and angular speed (per second) of a 10. cm second hand. Note: Each revolution generates radians. a.) Linear Speed: A second hand travels half the circumference in seconds: or b.) Angular Speed: Note: The angular speed does not depend on the length of the second hand! For eample: The riders on a carousel all have the same angular speed yet the riders on the outside have a greater linear speed than those on the inside due to a larger radius. Page 4 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
5 E5: Find the speed in mph of 36 in diameter wheels moving at 630 rpm (revolutions per minute). Unit Conversion: 1 statute (land) mile = 580 feet Earth s radius 3956 miles 1 nautical mile = 1 minute of arc length along the Earth s equator E6: How many statute miles are there in a nautical mile? s r r 3956 N Bearing: the course of an object given as the angle measured clockwise from due north E7: Use the picture to find the bearing of the ship N? You Try 1. A lawn roller with 10 inch radius wheels makes 1. revolutions/second. Find the linear and angular speed.. How many nautical miles are in a statute mile? Show your work. Reflection: QOD: How are radian and degree measures different? How are they similar? Page 5 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
6 Date: 4. Trigonometric Functions: The Unit Circle Syllabus Objectives: Syllabus Objectives: 3.1 The student will solve problems using the unit circle. 3. The student will solve problems using the inverse of trigonometric functions. The coordinates and y are two functions of the real variable t. You can use these coordinates to define the si trigonometric functions of t. sine cosecant cosine secant tangent cotangent Let t be a real number and let (,y) be the point on the unit circle corresponding to t. 1 sin t y csct y 1 cos t sec t= y tan t cot t y The Unit Circle: r 1 Page 6 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
7 Sin t Cos t Tan t Csc t Sec t Cot t E 1: Evaluate the si trigonometric functions at each real number. 5 a.) t b.) t c.) t 0 d.) t 6 4 You Try: Find the following without a calculator. 1. os. 5 cot 3 3. sin5 Periodic Functions: A function y f t f t c f t for all values of t in the domain of f. is periodic if there is a positive number c such that The period of sine and cosine are and the period of tangent is. E: Evaluate 3601 sin without a calculator. Eploration: Consider an angle, θ, and its opposite, as shown in the coordinate grid. Compare the trig functions of each angle. z θ θ z y y sin, sin cos, cos tan, tan csc, csc sec, sec cot, cot **Cosine and secant are the only EVEN f f trig functions. All the rest are ODD f f. Page 7 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
8 OddEven Identities: sin sin and csc csc tan tan and cot cot cos cos and sec sec E3: Find a.) sin (t) b.) cos (t) c.) Simplify the epression sin csc. Evaluating Trigonometric Ratios on the Calculator Note: Check the MODE on your calculator and be sure it is correct for the question asked (radian/degree). Unless an angle measure is shown with the degree symbol, assume the angle is in radians. E5: Evaluate the following using a calculator. a. sin4.68 Mode: degree sin b. sec1. Mode: radian Note: There is not a key for secant. We must use 1/(cos) because secant is the reciprocal of cosine. sec Also recall: csc and cot sin tan c. tan7 135 Mode: degree Inverse Trigonometric Functions: use these to find the angle when given a trig ratio E6: a.) Find θ (in degrees) if 5 cos. 1 cos Mode: degrees b.) sin t = 1/3 REFLECTION: How is the unit circle used to evaluate the trigonometric functions? Eplain. Page 8 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
9 Date: 4.3 Right Triangle Trigonometry Syllabus Objectives: 3.1 The student will solve problems using the unit circle. 3. The student will solve problems using the inverse of trigonometric functions. 3.5 The student will solve application problems involving triangles. Opposite Hypotenuse θ Adjacent Hypotenuse Adjacent Opposite Note: The adjacent and opposite sides are always the legs of the right triangle, and depend upon which angle is used. Trigonometric Ratios of θ: opposite Sine: sin Cosecant: hypotenuse Cosine: Tangent: adjacent cos Secant: hypotenuse opposite tan Cotangent: adjacent Memory Aid: SOHCAHTOA Reciprocal Functions hypotenuse csc opposite hypotenuse sec adjacent adjacent cot opposite Think About It: Which trigonometric ratios in a triangle must always be less than 1? Why? E1: For ABC, find the si trig ratios of A. sin A cos A tan A csc A sec A cot A Note: It may help to label the sides as Opp, Adj, and Hyp first. Find the si trig ratios for B. sin A cos A tan A csc A sec A cot A B 6 C 10 8 A Page 9 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
10 Special Right Triangles: & Formulate Chart. This table must be memorized! θ (Degrees) θ (Radians) sinθ cosθ tanθ 4.3 Identities Special Relationships: Identity: a statement that is true for all values for which both sides are defined Eample from algebra: Eploration: Consider a right triangle a Φ Note that b c θ and are complementary. Write the trig functions for each angle. What do you notice? sin cos cos sin tan cot csc sec sec csc cot tan **The trig functions of are equal to the cofunctions of θ, when and are complementary. Page 10 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
11 Cofunction Identities: sin 90 cos or sin cos cos 90 sin or cos sin sec 90 csc or sec csc csc 90 sec or csc sec tan 90 cot or cot tan cot 90 tan or cot tan E. a. sin 1 =. tan = c. sec 5 = E3. cos α =.8 Find sin α and tan α using identities. Reciprocal Identities sin cos tan csc sec cot csc sec cot sin cos tan Quotient Identities sin cos tan cot cos sin Recall: Unit Circle r 1, cos, y sin Note: sin sin y, Pythagorean Theorem: y 1 Pythagorean Identity: sin cos 1 To derive the other Pythagorean Identities, divide the entire equation by sin and then by cos : Page 11 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
12 Pythagorean Identities sin cos 1 1 cot csc tan 1 sec Simplifying Trigonometric Epressions: Look for identities Change everything to sine and cosine and reduce E4: Use basic identities to simplify the epressions. cos a) cot1 cos cot sin cos 1 sin 1 cos sin b) tancsc sin tan cos 1 csc sin E5: Simplify the epression csc 1 csc 1 Use algebra:. cos 1 cot csc cot csc 1 Application Problem E6: A 6 ft in man looks up at a 37 angle to the top of a building. He places his heel to toe 53 times and his shoe is 13 in. How tall is the building? y 37 74'' 74'' 53(13)=689'' Draw a picture: Use trig in the right triangle to solve for y: The height of the building is (y + 74) inches. Height of the building in inches = Convert to feet: The building is approimately tall. Page 1 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
13 q 3 N m You Try: Solve the triangle (find all missing sides and angles). M 8 Q Reflection QOD: Eplain how to find the inverse cotangent of an angle on the calculator. Page 13 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
14 Date: 4.4 Trigonometric Functions of Any Angle Syllabus Objectives: 3.1 The student will solve problems using the unit circle. 3. The student will solve problems using the inverse of trigonometric functions. Trigonometric Functions of any Angle II (S) sin (+) tan (+) III (T) r y θ cos (+) IV (C) I (A) sin (+) cos (+) tan (+) y r r y y sin r cos r y tan r csc y r sec cot y Review: Memory Aid: To remember which trig functions are positive in which quadrant, remember All Students Take Calculus. A all are positive in QI, S sine (and cosecant) is positive in QII, T tangent (and cotangent) is positive in QIII, C cosine (and secant) is positive in QIV. Review: Reference Angle: every angle has a reference angle, with initial side as the ais Have Memorized: Degrees 0 / / 3 Radians sinθ cosθ tanθ und 0 und 1. Find the Quadrant (Use: All Students Take Calculus). Find the Sign (positive or Negative) 3. Find Reference Angle 4. Evaluate (see chart above) *See timed test of Trigonometric Functions. Page 14 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
15 5 E1: Find the si trig functions if sin and tan 0. 7 Since both sine and tangent are positive, we know that θ is in the quadrant. Find the missing leg (a): a 5 7 a This is the adjacent leg of angle α. 5 opp sin 7 hyp y r 5 7 sin csc α cos sec tan cot E: Find the sin of 8 3 Solving a Triangle: find the missing angles and sides with given information A 3 5 E3: Solve the triangle (find all missing sides and angles). a 4 (Pythagorean Triple: 345; or use the Pythagorean Thm) C a B Page 15 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
16 Evaluating Trig Functions Given a Point Use the ordered pair as and y Find r: r y Check the signs of your answers by the quadrant E4: Find the si trig functions of θ in standard position whose terminal side contains point 5,3. Note: The point is in QII, so sine (and cosecant) will be positive. sin csc cos sec tan cot E5: Find the 6 trig functions of 330. To find the reference angle, start at the positive ais and go counterclockwise 330. Reference Angle =, y, r ( ) sin csc 3 1 cos sec tan cot Quadrantal Angles: angles with the terminal side on the aes, for eample, 0,90,180,70 Needs to be memorized! Degrees 0 / Radians sinθ cosθ tanθ 0 / E6: Find csc13. Find the reference angle: Reflection: Page 16 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
17 Date: 4.5 Graphs of Sine and Cosine Functions Syllabus Objectives: 4.1 The student will sketch the graphs of the si trigonometric functions. 4.3 The student will graph transformations of the basic trigonometric functions. 4.6 The student will model and solve realworld application problems involving sinusoidal functions. Teacher Note: Have students fill out the table as quickly as they can. Discuss patterns they can use to be able to memorize these. θ radians 0 sinθ cosθ tanθ Sinusoid: a function whose graph is a sine or cosine function; can be written in the form y asinb c d Sinusoidal Ais: the horizontal line that passes through the middle of a sinusoid EX1 Graph of the Sine Function Use the table above to sketch the graph. a.) Radians: y sin b.) Degrees: y sin Sine is periodic, so we can etend the graph to the left and right. Characteristics: Domain: Range: yintercept: intercepts:; Absolute Ma = Absolute Min = Decreasing: Increasing: Period = Sinusoidal Ais: Page 17 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
18 EX Graph of the Cosine Function Use the table above to sketch the graph. a.) Radians: y cos b.) Degrees: y cos Cosine is periodic, so we can etend the graph to the left and right. Characteristics: Domain: Range: yintercept: intercepts: Absolute Ma = Absolute Min = Decreasing: Increasing: Period = Sinusoidal Ais: Note: We will work mostly with the graphs in radians, since it is the graph of the function in terms of. Recall: f a h k is a transformation of the graph of f stretch/shrink, h is a horizontal shift and k is a vertical shift. Transformations of Sinusoids: General Form sin y a b h k a: vertical stretch and/or reflection over ais amplitude = a. a represents the vertical h: horizontal shift phase shift = h k: vertical shift period = b frequency = b = number of cycles completed in b Amplitude: the distance from the sinusoidal ais to the maimum value (half the height of a wave) E3: Sketch the graph of y 4sin. Vertical stretch: Amplitude = Page 18 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
19 Period: the length of time taken for one full cycle of the wave P b Frequency: the number of complete cycles the wave completes per unit of time E4: Sketch the graph of y sin. freq = b Period = Frequency = Reflection: If a 0, the sinusoid is reflected over the ais. E5: Sketch the graph of y cos. Note: Period, amplitude, etc. all stay the same. Vertical Translation: in the sinusoid sin y a b h k, the line y k is the sinusoidal ais E6: Sketch the graph of y4 sin. Sinusoidal Ais: y asin b h k Page 19 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter Phase Shift: the horizontal translation, h, of a sinusoid
20 E7: Sketch the graph of y cos. Shift y cos to the right. Does this graph look familiar? It is Note: Every cosine function can be written as a sine function using a phase shift. y cos y sin E8: Sketch the graph of y cos4. Then rewrite the function as a sine function. The coefficient of must equal 1. Factor out any other coefficient to find the actual horizontal shift. y cos4 y cos 4 4 Phase Shift: Period: Graph: Write as a sine function: y cos4 Writing the Equation of a Sinusoid E9: Write the equation of a sinusoid with amplitude 4 and period that passes through 6,0. 3 Amplitude: ; Period: b ; 3 b Phase Shift: normally passes through 0,0, so shift right 6 units Page 0 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
21 You Try: Describe the transformations of y 7sin 0.5. Then sketch the graph. 4 Reflection QOD: How do you convert from a cosine function to a sine function? Eplain. Page 1 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
22 Date: 4.6 Graphs of Other Trigonmetric Functions Syllabus Objectives: 4.1 The student will sketch the graphs of the si trigonometric functions. 4.3 The student will graph transformations of the basic trigonometric functions. θ radians 0 sinθ cosθ tanθ E 1 Graph of the Tangent Function: Use the table above to sketch the graph. a.) Radians: y tan b) Degrees: y tan Characteristics: Domain: Range: Intercepts: Increasing/Decreasing: Period: Vertical Asymptotes: sin Note: tan, so the zeros of sine are the zeros of tangent, and the zeros of cosine are the vertical cos asymptotes of tangent. E Graph of the Cotangent Function Use the table above to sketch the graph. (Cotangent is the reciprocal of tangent.) a.) Radians: y cot b.) Degrees: y cot Characteristics: Domain: Range: Intercepts: Increasing/Decreasing Period: Vertical Asymptotes: cos Note: cot, so the zeros of cosine are the zeros of cotangent, and the zeros of sine are the vertical sin asymptotes of cotangent. Page of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
23 E. 3 Graph of the Secant Function Use the table above to sketch the graph. (Secant is the reciprocal of cosine.) a.) Radians: y sec b.) Degrees: y sec Note: It may help to graph the cosine function first. Characteristics: Domain: Range: Intercept: Local Ma: Local Min: Period: Vertical Asymptotes: E. 4 Graph of the Cosecant Function Use the table above to sketch the graph. (Cosecant is the reciprocal of sine.) a.) Radians: y csc b.) Degrees: y csc Note: It may help to graph the sine function first. Characteristics: Domain: Range: Intercept: Local Ma: Local Min: Period: Vertical Asymptotes: Transformations: Sketch the graph of the reciprocal function first! E5: Sketch the graph of y1 sec. Graph y1 cos. Shift up 1, amplitude. Use the zeros of the cosine function to determine the vertical asymptotes of the secant function. Sketch the secant function. Note: Amplitude is not applicable to secant. The number represents a vertical stretch. Page 3 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
24 E6: Sketch the graph of y tan. Period: P Reflect over ais 1 Solving a Trigonometric Equation Algebraically E7: Solve for in the given interval using reference triangles in the proper quadrant. sec, Quadrant II; cos Reference Angle = In QII: Solving a Trigonometric Equation Graphically E8: Use a calculator to solve for in the given interval: cot 5, 0 1 The equation cot 5 is equivalent to the equation tan. Graph each side of the equation and find 5 the point of intersection for 0. Restrict the window to only include these values of. 1 Note: We could have also graphed y1 and y 5. (Shown in the second graph above.) tan Solution: You Try: Graph the function y csc. Reflection: QOD: 1. What trigonometric function is the slope of the terminal side of an angle in standard position? Eplain your answer using the unit circle.. Which trigonometric function(s) are odd? Which are even? Page 4 of 4 Precalculus Graphical, Numerical, Algebraic: Larson Chapter
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