Trigonometric Functions

Trigonometric Functions 015 College Board. All rights reserved. Unit Overview In this unit ou will build on our understanding of right triangle trigonometr as ou stud angles in radian measure, trigonometric functions, and circular functions. You will investigate in depth the graphs of the sine and cosine functions and etend our knowledge of trigonometr to include tangent, cotangent, secant, and cosecant, as well as solving trigonometric equations. Ke Terms As ou stud this unit, add these and other terms to our math notebook. Include in our notes our prior knowledge of each word, as well as our eperiences in using the word in different mathematical eamples. If needed, ask for help in pronouncing new words and add information on pronunciation to our math notebook. It is important that ou learn new terms and use them correctl in our class discussions and in our problem solutions. Academic Vocabular subtend Math Terms initial side terminal side standard position coterminal angles subtend radian angular velocit linear velocit reference triangle periodic function period amplitude phase shift trigonometric functions sine unit circle tangent cosecant secant tangent concentric circles one-to-one function inverse trigonometric function reference angle ESSENTIAL QUESTIONS What tpe of real-world problems are modeled and solved using trigonometr? How are graphic representations of trigonometric functions useful in understanding real-life phenomena? EMBEDDED ASSESSMENTS This unit has two embedded assessments, following Activities 18 and 0. These assessments will give ou an opportunit to demonstrate what ou have learned about trigonometric functions and their inverses, the graphs of sinusoidal curves, and writing and solving trigonometric equations. Embedded Assessment 1: Angles, the Unit Circle, and Trigonometric Graphs p. 5 Embedded Assessment : Inverse Trigonometric Functions and Trigonometric Equations p. 75 185

UNIT Getting Read Write our answers on notebook paper. Show our work. 1. Find the measure of each angle of triangle ABC, given AC =, CB =, and AB =.. Which equation could ou use to find the measure of in the figure below? 8 90 A. sin = 8 B. cos = 8 C. tan = 8 D. sin = 8 E. cos = 8. In DEF, m D = 70, m E = 90, and EF = 10. Find, to the nearest whole number, the measure of the hpotenuse of DEF.. In GHI, GH = 1 ft, HI = 11 ft, and m H = 90. Find the measure of G and the length of GI. 5. Eplain this statement: The inverse of a function is sometimes a function. Use eamples in our eplanation.. Consider the function f( ) = 1. a. Give the domain and the range. b. Sketch a graph of the function. 10 8 10 8 8 10 8 10 c. Write the inverse function. d. Give the domain and the range of the inverse function. e. Sketch a graph of the inverse function on the same coordinate grid as in Item b. 7. Determine whether f( ) = + 1 and g() = are inverse functions. Justif our answer. 8. Eplain how the graph of = ( 1) + differs from the graph of =. Eplain how ou can determine the differences without graphing. 015 College Board. All rights reserved. 18 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Angles and Angle Measure What s M Angle Measure? Lesson 1-1 Angle Measures in Standard Position Learning Targets: Draw angles in standard position. Find the initial side and terminal side of an angle in standard position. Identif coterminal angles. SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Summarizing, Paraphrasing, Vocabular Organizer, Create Representations, Think-Pair-Share In geometr, an angle has two fied ras with a common endpoint. 1. Attend to precision. Draw an eample of each of the following tpes of angles, and give the degree measure of the angle. Then state the possible range of measures for all angles of that tpe. a. acute angle b. obtuse angle ACTIVITY 1 c. right angle d. straight angle 015 College Board. All rights reserved. In trigonometr, an angle consists of a fied ra, called the initial side, and a rotating ra, called the terminal side. An angle is in standard position when the verte is at the origin and the initial side is on the positive -ais. A counterclockwise rotation represents an angle with positive measure. A clockwise rotation represents an angle with negative measure. O Terminal side Initial side Terminal side O Initial side. Draw each angle described below in standard position, and then give the degree measure of the angle. a. one-fourth of a complete counterclockwise rotation b. one-half of a complete clockwise rotation c. one complete counterclockwise rotation (the angle formed b rotating the initial side counterclockwise until it coincides with itself) d. one-eighth of a complete clockwise rotation e. two and one-third complete counterclockwise rotations CONNECT TO HISTORY Dividing a circle into 0 parts can be traced to the ancient cit of Bablon. The Bablonians used a base-0 number sstem instead of the base-10 sstem we use toda. Activit 1 Angles and Angle Measure 187

ACTIVITY 1 Lesson 1-1 Angle Measures in Standard Position MATH TERMS The 15 angle and the 5 angle shown below are formed b different rotations, et the have the same initial and terminal sides. Therefore, the are coterminal angles. Coterminal angles are angles formed b different rotations but with the same initial and terminal sides. To find measures of coterminal angles, add or subtract multiples of 0.. Make use of structure. For each angle in standard position, find two positive angles and two negative angles that are coterminal with the given angle. a. 0 b. 100 5 O 15. Find an angle between 0 and 0 that is coterminal with the given angle. a. 890 b. 150 The circumference of a circle is equal to πr, where r is the radius of the circle. The length of an arc subtended b a central angle of n degrees is equal to n r 0 ( π ). MATH TERMS n (πr) 0 When the sides of an angle pass through the endpoints of an arc, the angle subtends the arc. In the figure below, the arc PQ is subtended b angle POQ. P n r O Q Eample A A pet gerbil runs the length of an arc on a circular wheel with radius 1.5 cm which subtends a central angle of 0. What length did the gerbil travel? 0 0 ( π ( 1. 5) ) Substitute the radius and angle measure. 8.7 cm Simplif. Tr These A Find the length of each arc to the nearest hundredth of a unit. a. b. 0 8 in. 10 0 cm 015 College Board. All rights reserved. 5. Make sense of problems. A wheel with a radius of 0 inches rotated 80 as it rolled along the ground. How far did the wheel travel? 188 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1-1 Angle Measures in Standard Position ACTIVITY 1. The second hand of a clock is inches long. a. How man degrees does the second hand rotate in 1 minutes? CONNECT TO GEOGRAPHY b. How far does the tip of the second hand travel in 1 minutes? 7. What is the length of the major arc AB? A nautical mile is approimatel equal to the length of a 1 0 -degree arc on a great circle of the earth line of longitude. B A Check Your Understanding 8. Epress regularit in repeated reasoning. Write an epression that generates ever angle coterminal with a 78 angle. 9. Which arc has a greater length: an arc subtended b a 0 angle on a circle with radius 1 cm or an arc subtended b a 5 angle on a circle with radius 1 cm? 015 College Board. All rights reserved. LESSON 1-1 PRACTICE 10. An angle of 91 is drawn on a coordinate plane with its verte at the origin and its initial side on the positive -ais. In which quadrant does the terminal side lie? 11. Find an angle between 0 and 0 that is coterminal with the given angles. a. 110 b. 0 1. What is the perimeter of each figure? a. b. 18 9 mm 15 inches 1. The end of a pendulum with a length of 1 cm travels an arc length of cm. How man degrees does the pendulum swing? 1. Critique the reasoning of others. Neil believes the two arcs subtended b a 195-degree angle and b a 15-degree angle in a circle with a radius of 10 inches will have the same length. Eplain Neil s mistake. Activit 1 Angles and Angle Measure 189

ACTIVITY 1 Lesson 1- Radian Measure Learning Targets: Measure angles in radians. Convert angle measures from degrees to radians. Recognize trigonometric ratios to complete reference triangles. SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Create Representations, Vocabular Organizer, Think-Pair-Share, Group Presentations Degrees are not the onl unit of measure for angles. To find out about another unit of angle measure, complete the following questions concerning circles A and B, shown below. Circle A Circle B 1. Cut a piece of string equal in length to the radius of each circle. a. Bend the string around each circle and mark off as man arcs as possible that are equal in length to the radii. Write the approimate number of radii that fit around each circle. Circle A: Circle B: b. Reason quantitativel. What do ou notice about our answers in part a? How can ou eplain this? What is the eact number of radii that fit around each circle? 015 College Board. All rights reserved. c. In each circle, mark off a central angle that intercepts an arc with the same length as the radius of the circle. 190 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1- Radian Measure ACTIVITY 1 Angles ma be measured in radians as well as in degrees. One radian is the measure of a central angle which intersects an arc equal in length to the radii. O r r A r B MATH TERMS A central angle of a circle which subtends an arc equal to the length of the radius of the circle has a measure of one radian. m AOB = 1 radian CONNECT TO AP. Suppose that angle θ is in standard position and formed b one complete counterclockwise rotation. a. What is the measure of angle θ in degrees? In calculus, angles are measured in radians, since proofs of major theorems are based on radian measure. b. What is the measure of angle θ in radians? Give an eact answer in terms of π.. Use our answers to Item to complete the following. A full rotation = degrees = radians 1 degree = radians 1 radian = degrees WRITING MATH Degree measure is denoted using the degree smbol ( ), while radian measure is written without an smbol. 015 College Board. All rights reserved.. Convert each degree measure to radians. Give eact answers. a. 5 b. 10 c. 15 5. Convert each radian measure to the nearest degree. a. π b. π c.. Name the quadrant or ais where the terminal side of each angle lies. a. π b. 5 7π c. π d. 11 8π MATH TIP When the terminal side of an angle lies in a certain quadrant or on a certain ais, we sa that the angle lies in that quadrant or on that ais. Activit 1 Angles and Angle Measure 191

ACTIVITY 1 Lesson 1- Radian Measure 7. Draw each angle in standard position. Write the radian measure. a. one-fourth of a complete clockwise rotation b. two-thirds of a complete counterclockwise rotation c. one and three-fourths counterclockwise rotations 8. Epress regularit in repeated reasoning. Eplain how to find angles coterminal to a given angle measured in radians. 9. For each angle in standard position, find one positive angle and one negative angle that is coterminal with the given angle. a. π 5 b. π 10. Find an angle between 0 and π that is coterminal with each angle. 015 College Board. All rights reserved. a. 1π b. 5π 19 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1- Radian Measure ACTIVITY 1 An object rotating about a point has both angular velocit and linear velocit. Angular velocit is the rate of change in angle measure as the object rotates. Linear velocit is the rate at which location changes as the object rotates. Eample A A Ferris wheel with a radius of 5 feet rotates at a speed of.5 revolutions per minute (rpm). Find the angular velocit, in radians per minute, and the linear velocit, in miles per hour, of a point on the outer edge of the Ferris wheel. Angular velocit is the ratio of radians rotated to time.. 5( π) = 5π Find the number of radians in.5 revolutions. 5π radians Epress angular velocit as a ratio. 1 minute Linear velocit is the ratio of feet traveled to time in hours.. 5( π( 5) ) 70. 8 feet Find the approimate distance traveled b multipling the number of revolutions b the circumference. 70.8 feet Epress linear velocit as a ratio. 1 minute 70. 8 feet 1 0 1 minute mile minutes Multipl to change the units. 580 feet 1 hour 8.0 miles per hour Simplif. MATH TERMS Angular velocit, w, is equal to the measure, in radians, of the angle of rotation divided b time: w = θ. Linear velocit, v, is equal t to the length of the arc subtended b the angle of rotation divided b time: v = θ r. t MATH TIP 1 mile = 5,80 feet Tr These A a. Find the linear velocit, in centimeters per second, of an object rotating at 9 rpm around a point 1 centimeters awa. b. Reason quantitativel. A biccle is ridden at a constant speed of miles per hour. What is the angular velocit, in radians per second, of its wheels if the have a diameter of inches? 015 College Board. All rights reserved. 11. The propeller blades of a single-engine aircraft rotate at a speed of,0 rpm at takeoff. What is the angular velocit of the blades in radians per second? 1. Two pulles, one with a radius of inches and the other with a radius of 9 inches, are connected b a belt. The larger pulle has an angular velocit of 0 rpm. DISCUSSION GROUP TIPS If ou do not understand something in group discussions, ask for help or raise our hand for help. Describe our questions as clearl as possible, using snonms or other words when ou do not know the precise words to use. a. What is the linear speed of the belt in feet per second? b. What is the angular velocit of the smaller pulle in revolutions per minute? 1. The second hand of a clock is 5 inches long. What is the linear speed of the tip of the second hand in feet per hour? Activit 1 Angles and Angle Measure 19

ACTIVITY 1 Lesson 1- Radian Measure Given an angle in standard position, a reference triangle is formed b drawing a perpendicular segment from a point on the terminal side of the angle to the -ais. 1. Each figure below shows an angle θ in standard position. Draw a perpendicular segment from point P to the -ais to form a reference triangle. a. b. P P θ O θ O c. d. O θ O θ P P 15. Draw a reference triangle and find the missing value for each figure. a. Find OP. b. Find OP. P(, ) P(, ) θ O θ O c. OP = ; Find. d. OP = 10; Find. θ θ O O P(, 5) P(7, ) 015 College Board. All rights reserved. 19 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1- Radian Measure ACTIVITY 1 Recall the following right triangle trigonometric ratios. opposite leg adjacent leg sin θ = cos θ = hpotenuse hpotenuse opposite leg tan θ = adjacent leg csc θ = hpotenuse opposite leg adjacent leg cot θ = opposite leg sec θ = hpotenuse adjacent leg 1. Given that θ is an angle in standard position, O is the origin, and P is a point on the terminal side of θ. For each of the following, draw a figure with a reference triangle. Then find the missing value, correct to three decimal places. a. P(, ); θ = 70 ; Find. Opposite leg Hpotenuse θ Adjacent leg TECHNOLOGY TIP Most calculators can evaluate trigonometric functions for angle measures in either degrees or radians. Be sure to have our calculator in the correct mode when solving problems. b. OP = 0; θ = 8 ; Find the coordinates of point P. CONNECT TO HISTORY 015 College Board. All rights reserved. c. P(, ); Find θ. Although the stud of trigonometric ratios dates back more than,000 ears, the term trigonometr was coined in 1595 and is derived from the Greek word trigonometria, meaning triangle measuring. d. OP = 1; θ = 0 ; Find the coordinates of point P. Activit 1 Angles and Angle Measure 195

ACTIVITY 1 Lesson 1- Radian Measure Check Your Understanding 17. Eplain how to find each coordinate of P. P 1 75 a. the -coordinate of P b. the -coordinate of P 18. State two angles in radians, between π and π, that are coterminal with a 75 angle. LESSON 1- PRACTICE 19. An angle of 5 π radians is drawn on a coordinate plane, with its verte at 9 the origin and its initial side on the positive -ais. In which quadrant does the terminal side lie? 0. Find an angle between 0 and π that is coterminal with the given angles. a. π b. π c. π 9 1. The blade of a circular saw has a radius of inches. The blade spins at 5,00 rpm. a. What is the angular velocit of the blade in radians per second? b. What is the linear velocit of the teeth at the end of the blade in miles per hour?. Find the coordinates of P to three decimal places. P 1. Critique the reasoning of others. Heather believes OP = 11. Eplain Heather s mistake. 015 College Board. All rights reserved. ( 5, ) P θ O 19 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Angles and Angle Measure What s M Angle Measure? ACTIVITY 1 ACTIVITY 1 PRACTICE Write our answers on notebook paper. Show our work. Lesson 1-1 1. Draw one-third of a complete counterclockwise rotation as an angle in standard position, and then give the measure of the angle in degrees.. Draw two and one-eighth complete clockwise rotations as an angle in standard position, and then give the measure of the angle in degrees.. Find an angle between 0 and 0 that is coterminal with a 905 angle.. Find an angle between 0 and 0 that is coterminal with a angle. 5. Which pair of angles is coterminal? A. 00 and 00 B. 50 and 50 C. 80 and 80 D. 70 and 70. An angle of 1 is drawn on a coordinate plane, with its verte at the origin and its initial side on the positive -ais. In which quadrant does the terminal side lie? 7. Find the perimeter of the shaded region in the figure below, correct to three decimal places. 8. What is the value of in the figure below, correct to three decimal places? cm cm 9. The end of a pendulum with a length of 1 inches travels an arc length of 10 inches. How man degrees, correct to three decimal places, does the pendulum swing? Lessons 1- and 1-10. Convert each degree measure to radians. Give eact answers in terms of π. a. 150 b. 0 c. 1,090 11. Convert each radian measure to degrees. a. π b. 19 1 π c. 7 10 π 1. Draw three-eighths of a complete counterclockwise rotation as an angle in standard position, and then give the measure of the angle in radians. 015 College Board. All rights reserved. 15 10 mm Activit 1 Angles and Angle Measure 197

ACTIVITY 1 Angles and Angle Measure What s M Angle Measure? 1. Draw one and two-thirds of a complete counterclockwise rotation as an angle in standard position, and then give the measure of the angle in radians. 1. Find an angle between 0 and π that is coterminal with an angle of 1 7π radians. 15. Find an angle between 0 and π that is coterminal with an angle of 15 11π radians. 1. A washing machine drum rotates at a speed of 1,500 rpm during the spin ccle. What is the angular velocit of the drum in radians per second? A. 5 π radians per second B. 50π radians per second C. 5,000π radians per second D. 180,000π radians per second 17. A conveor belt rolls on clindrical bearings that have a radius of 1. inches. The belt has a velocit of. inches per minute. Find the angular velocit of the roller bearings in radians per minute, correct to three decimal places. 18. A pulle 8 inches in diameter rotates with an angular velocit of 150 rpm. Find the linear velocit, in feet per second, of a cable attached to the pulle, correct to three decimal places. 19. Draw a reference triangle and find OP. 0. Let O be the origin and P be a point on the terminal side of θ, an angle in standard position. Find OP when P is (, 11). 1. Let O be the origin and P be a point on the terminal side of θ, an angle in standard position. Find the coordinates of P when OP = 0 and θ =, correct to three decimal places.. Let O be the origin and P be a point on the terminal side of θ, an angle in standard position. Find OP when the -coordinate of P is 1 and θ = 50, correct to three decimal places.. Let O be the origin and P be a point on the terminal side of θ, an angle in standard position. Find θ in degrees when P(10, 10). MATHEMATICAL PRACTICES Reason Abstractl and Quantitativel. Two groups of students built catapults in their science classes. The catapults were identical in design with one eception. In both catapults, the arm of the catapult rotated about a fied point to launch the projectile, and in both catapults, the arm rotated through the same measure arc with the same angular velocit. However, the arm of Group A s catapult was inches longer than the arm of Group B s catapult. Which catapult launches items with the greater linear speed? Eplain wh this is true. O θ P(7, 9 ) 015 College Board. All rights reserved. 198 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Sinusoidal Functions Biccle Wheels Lesson 15-1 Eploring Periodic Data ACTIVITY 15 Learning Targets: Recognize situations that involve periodic data. Sketch a graph of periodic data. SUGGESTED LEARNING STRATEGIES: Summarizing, Paraphrasing, Create Representations, Visualization, Use Manipulatives, Group Presentation, Quickwrite As ou read, mark the tet to identif ke information and parts of sentences that help ou make meaning from the tet. Stac has a new bike. The bike has -inch-diameter wheels and -inchdiameter training wheels. The horizontal distance between the center of the -inch front wheel and the center of one -inch training wheel is inches. Stac is riding at a stead pace, and the -inch wheels rotate once ever seconds. As Stac is riding down the street, her bike runs over a freshl painted parking stripe, and each wheel picks up a narrow strip of fresh paint that leaves marks on the pavement. MATH TIP In one rotation, a wheel will travel a distance equal to the length of the circumference of the wheel. 1. Model with mathematics. The figure at the right represents the front wheel and a training wheel on Stac s bike. Label the length of each of the three segments shown in the figure, and then summarizean additional information given in the opening paragraph. 015 College Board. All rights reserved.. Let t = 0 seconds represent the time when Stac s front wheel first crosses the freshl painted stripe. Sketch a graph of the height above the pavement of the paint spot on the front wheel as a function of the number of seconds for the first 8 seconds after t = 0. inches t seconds. Assume that Stac s biccle is on a path that runs perpendicular to the paint stripe. a. Find the distance that the front wheel travels in seconds. Then use this information to find how long it takes for the training wheel to make one complete revolution. Activit 15 Sinusoidal Functions 199

ACTIVITY 15 Lesson 15-1 Eploring Periodic Data b. How man seconds will it take from t = 0 seconds until the training wheel first runs over the freshl painted stripe? c. On the grid in Item, sketch a graph of the height of the paint spot on the training wheel as a function of the number of seconds elapsed since t = 0.. Use the sketch from Item c to estimate the first time that the paint spots on the front wheel and on the training wheel will be eactl the same height above the pavement. 5. A more accurate graph would give a better approimation of the time ou found in Item. a. How man seconds after hitting the wet stripe will the paint spot first be in this position on the front wheel? 10 Paint b. Make use of structure. Eplain how special right triangles can be used to find the height of the paint spot. c. Find the height of the paint spot to the nearest tenth of an inch. d. Is there an other instance during the first revolution of the wheel where the paint spot is at this same height? How long after hitting the stripe does this happen? 015 College Board. All rights reserved. 00 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 15-1 Eploring Periodic Data ACTIVITY 15 e. For each value of t, draw a central angle with a vertical initial side showing the position of the paint spot and the angle of rotation since passing the stripe. Then determine the height of the paint spot to the nearest tenth of an inch. Assume the wheel spins counterclockwise as Stac rides. t = 1 sec t = sec t = 1 sec MATH TIP Special Right Triangles 0 0 90 1 t = 1 sec t = 1 sec t = sec 0 0 5 5 90 5 1 t = sec t = sec t = sec 5 015 College Board. All rights reserved. t = 1 sec t = sec t = sec Activit 15 Sinusoidal Functions 01

ACTIVITY 15 Lesson 15-1 Eploring Periodic Data Check Your Understanding. Epress regularit in repeated reasoning. Find three other instances in the first 0 seconds after passing the stripe when the paint spot is in the same position on Stac s wheel as it is when t = seconds. Eplain how ou know this. 7. Critique the reasoning of others. Wh would a graph with straight sides be an inappropriate answer to Item? MATH TIP The height of the center of each circle is half the diameter. The height of each paint spot can be found b adding or subtracting the length of a side of a 0-0 -90 right triangle from that measure. LESSON 15-1 PRACTICE 8. What is the height, to the nearest tenth of an inch, of the paint spot on the training wheel at the following times? Justif our answers. a. 1 second after the training wheel crosses the stripe b. 1 second after the training wheel crosses the stripe c. 1 11 second after the training wheel crosses the stripe d. 1 second after the training wheel crosses the stripe 8 e. 1 seconds after the training wheel crosses the stripe 015 College Board. All rights reserved. 0 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 15- Periodic Functions ACTIVITY 15 Learning Targets: Eplore how a change in parameters affects a graph. Determine the period, amplitude, or phase shift of a periodic function. SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite, Think-Pair-Share 1. Recall that Stac has a bike with -inch-diameter wheels and -inchdiameter training wheels. The horizontal distance between the center of the -inch front wheel and the center of one -inch training wheel is inches, and the -inch wheels rotate once ever four seconds. t represents the time, in seconds, since Stac s front wheel first crossed a freshl painted stripe of wet paint, making a spot on her wheel. a. Use the results from Item 5e in Lesson 15-1 and the aes below to construct a detailed graph of the height above the pavement of the paint spot on the front wheel as a function of the number of seconds that have elapsed since time t = 0 seconds for the first 8 seconds. 7 1 18 15 1 9 inches 1 5 7 8 t seconds 015 College Board. All rights reserved. b. Describe our graph. How does it compare to our initial graph from Item 1 in Lesson 15-1? c. If the graph were etended to include the first minute elapsed since t = 0, describe what the graph would look like. d. On the aes in part a, construct a graph for the training wheel. Include heights of the paint spot on the training wheel at 1 1 -second intervals for the first seconds after the training wheel tire first picks up the paint. e. Use the graph in part a to approimate the first time the paint spots on the front wheel and on the training wheel will be eactl the same height above the pavement. Activit 15 Sinusoidal Functions 0

ACTIVITY 15 Lesson 15- Periodic Functions Now we will eplore how changes in parameters affect the graph of the height of the paint spot on the front wheel.. Review our work from Item 5 in Lesson 15-1. a. Redraw the graph for the height of the paint spot on the -inch front wheel from Item 1 on the aes below. Suppose that the front wheel of Stac s bike had a diameter of 18 inches inches instead of but still rotated once ever seconds. Eplain how this would change the graph, and then sketch it on the same aes. 7 1 18 15 1 9 inches 1 5 7 8 t seconds b. Redraw the original graph for the height of the paint spot on the -inch front wheel from Item 1 on the aes below. Suppose that the wheel size remained inches but the rotational velocit was one revolution ever seconds instead of ever seconds. Eplain how the graph for the front wheel would change, and then sketch it on the same aes. 7 1 18 15 1 9 inches 1 5 7 8 t seconds 015 College Board. All rights reserved. 0 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 15- Periodic Functions ACTIVITY 15. How would the original graphs for both wheels have differed if the heights of the paint marks on the tires had been measured as a vertical distance above or below a line through the center of the -inch wheel and parallel to the ground? Eplain, and make a sketch to illustrate our answer. 015 College Board. All rights reserved.. How would the original graphs for both wheels have differed if the heights of the paint marks on the tires had been plotted as a distance above the pavement and as a function of time elapsed since the training wheel first crossed the fresh paint? Eplain, and make a sketch to illustrate our answer. inches 7 1 18 15 1 9 1 5 7 8 t seconds MATH TIP The time it takes the training wheel to travel d inches is equal to d π seconds. Activit 15 Sinusoidal Functions 05

ACTIVITY 15 Lesson 15- Periodic Functions DISCUSSION GROUP TIPS As our share our ideas, be sure to use mathematical terms and academic vocabular precisel. Ask our teacher if ou need help to epress our ideas to the group or class. A function f is a periodic function if there is a positive number p such that f() = f( + p) for all in the domain of f. The number p is the period of the function. If a periodic function has a maimum and minimum value, then the amplitude of the function is half the difference of the maimum and the minimum values. A phase shift is a horizontal translation of a periodic function. 5. Consider the functions that model the height of the paint spot on the front wheel. a. Eplain wh the function that models the original situation with the -inch wheel is periodic, and give the period of the function. Then tell which of the situations in Items changed the period of the function, and give the new value of the period. b. Give the amplitude of the function that models the original situation with the -inch wheel. Then tell which of the situations in Items changed the amplitude of the function, and give the new value of the amplitude. c. Tell which of the situations in Items caused a vertical translation of a function. Then describe the translation. d. Tell which of the situations in Items caused a phase shift of a function. Then describe the shift. Check Your Understanding. The graph of g() is shown below. Identif the amplitude and period of the function. 015 College Board. All rights reserved. 8 10 1 1 1 18 7. Let h() = h( + 10) for all in the domain of h. What does this impl about the function h()? 0 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 15- Periodic Functions ACTIVITY 15 LESSON 15- PRACTICE 8. Given the graph of f() below, state the period and amplitude of f(). 9. Let p() be a vertical translation up 1 unit of f() from Item 8. a. Sketch p(). b. What is the period of p()? c. What is the amplitude of p()? 10. Use appropriate tools strategicall. Marcus is using a graphing calculator to stud k(), a periodic function centered verticall about the line = 5. The period of k() is 8, and the amplitude is. The settings of his viewing window are shown below. What changes should Marcus make to the viewing window settings in order to better stud the function? Eplain. 8 10 CONNECT TO TECHNOLOGY In the window settings, Xmin is the minimum value displaed on the -ais. Xma is the maimum value displaed on the -ais. Xscl is the distance between the tick marks on the -ais. Ymin is the minimum value displaed on the -ais. Yma is the maimum value displaed on the -ais. Yscl is the distance between the tick marks on the -ais. Xres is the piel resolution. WINDOW Xmin= Xma=17 Xsc1=1 Ymin= 10 Yma=10 Ysc1=1 Xres=1 015 College Board. All rights reserved. Activit 15 Sinusoidal Functions 07

ACTIVITY 15 Lesson 15- Graphs and the Sine Function MATH TIP For the periodic function = f() with period p, ou know that f() = f( + p) for all in the domain of f. Furthermore, if = c + kp, where k is an integer, then f() = f(c + kp) = f(c). Learning Targets: Graph a periodic function with various domains. Compare the graph of = sin to periodic graphs. SUGGESTED LEARNING STRATEGIES: Identif a Subtask, Create Representations, Think-Pair-Share, Quickwrite 1. Use the heights of the paint spots that ou found in Item 5e in Lesson 15-1 and the fact that the function that models the original situation with the -inch wheel is periodic to find the height of the paint spot on the front wheel at each of the following times. Justif our answers. a. 1 seconds b. 1 seconds c. 19 seconds d. 8 seconds e. 9 1 seconds. Suppose that the height of the paint spot is measured as a vertical distance above or below the center of the -inch wheel and that the paint mark starts at a point on the same horizontal line as the center of the wheel at t = 0. Suppose also that the wheel turns in the direction shown b the arrow in the figure at the same rate as before (one revolution in seconds). Draw a graph of the height of the spot as a function of time for 0 t 8. inches 1 9 0 1 5 7 8 9 1 Paint spot when t = 0 Spoke s t seconds 015 College Board. All rights reserved. 08 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 15- Graphs and the Sine Function ACTIVITY 15. What happens if the unit of measure is feet rather than inches? Cop the graph in Item and label the aes to illustrate the change. 1 5 7 8 t seconds. Instead of defining the function as height versus time, consider defining it as the height of the paint spot in feet versus the angle of rotation, measured in degrees, of spoke s. Through how man degrees will the spoke rotate in 8 seconds? Cop the graph in Item and label both aes to reflect the change. 015 College Board. All rights reserved. 5. Cop the graph from Item and label the aes so that the graph illustrates the height of the paint spot, in feet, as a function of the angle of rotation of spoke s, measured in radians. Activit 15 Sinusoidal Functions 09

ACTIVITY 15 Lesson 15- Graphs and the Sine Function MATH TERMS The functions = sin (), = cos (), and = tan () are eamples of trigonometric functions. The sine function is defined as follows: sin θ = r, where + = r.. Put our graphing calculator in degree mode and set the window to match the graph in Item. a. Graph the function = sin() and compare the graph on our calculator to the graph in Item. b. In a right triangle in which θ is an acute angle, sin θ is defined as the length of opposite leg ratio. Eplain how this definition applies to length of hpotenuse our graph. P(, ) r θ O 7. Put our graphing calculator in radian mode and set the window to match the graph in Item 5. a. Graph the function = sin() and compare the graph on our calculator to the graph in Item 5. b. Identif the amplitude and period of the function = sin(). c. If P(, ) is an point on the terminal side of an angle θ in standard position, then the sine of θ is defined as sin θ = r, where + = r. Eplain how this definition of sine applies to our graph. 015 College Board. All rights reserved. 10 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 15- Graphs and the Sine Function ACTIVITY 15 Check Your Understanding 8. Epress regularit in repeated reasoning. Suppose = g() is a periodic trigonometric function with a period of 5π radians and amplitude of π feet when is measured in radians. What would be the period and amplitude of g() if were measured in degrees and were measured in ards? 10 5 0 15 10 5 5 5 10 15 0 10 015 College Board. All rights reserved. LESSON 15- PRACTICE 9. Reason quantitativel. Jenna created the periodic function = h() to model the depth of the water at the local marina during high and low tide. In Jenna s function, is the number of hours that have passed since midnight, and is the number of inches the water level at the marina is above or below 8 inches. Describe the effect each of the following changes has on the graph of = h(). a. is measured in minutes instead of hours. b. is the number of hours before or after p.m. instead of after midnight. c. is measured in feet instead of inches. d. is the depth of the water instead of the difference from 8 inches. MATH TIP 1 radian = 180 π degrees Activit 15 Sinusoidal Functions 11

ACTIVITY 15 Sinusoidal Functions Biccle Wheels ACTIVITY 15 PRACTICE Write our answers on notebook paper. Show our work. Lesson 15-1 Use this information for Items 1 1. A waterwheel has a diameter of feet. The center of the wheel is feet below the edge of a flume, and the wheel rotates at a stead rate of 10 revolutions per minute. Wheel Rotation Water level ft ft Broken Blade 1. How long does it take the wheel to complete one revolution?. One of the blades of the waterwheel is broken. Let = d(t) be a function relating, the vertical distance of the broken blade above or below the bottom of the flume, to time, t. Let t = 0 seconds represent the time when the broken blade is first touching the flume wheel. Sketch a graph of = d(t) for the first 0 seconds after t = 0.. What is the vertical distance the broken blade is from the bottom of the flume, correct to three decimal places, when t = seconds?. What is the vertical distance the broken blade is from the bottom of the flume, correct to three decimal places, when t = 5 seconds? 5. At what value of t is the broken blade of the wheel first at the point farthest from the flume?. At which two instances is the broken blade in the same position in its rotation? A. t = 11 and t = 1 B. t = 11 and t = 19 C. t = 11 and t = 1 D. t = 11 and t = Lesson 15-7. Is = d(t) a periodic function? 8. What is the amplitude of = d(t)? 9. If were changed to be the vertical distance of the broken blade above or below the center of the wheel, how would the graph of differ from the original graph in Item? 10. If the wheel slows to a stead rate of revolutions per minute, how would the graph of differ from the original graph in Item? 11. Which change in the scenario would result in a phase shift of the original function? A. changing the definition of t = 0 B. changing the units of t C. using a faster wheel D. using a larger wheel Lesson 15-1. Give the vertical distance from the bottom of the flume to the broken blade, correct to three decimal places, when t = 1 seconds. 1. Does this graph represent a periodic function? 5 1 7 5 0 1 1 1 5 7 1. Suppose that the graph in Item 1 is a graph of = f(). Find f() and justif our answer. 15. Sketch an eample of a periodic function with a period of 8 and amplitude of. 1. P is a point on the terminal side of a 50 angle in standard position. Find the coordinates of P, correct to three decimal places, given + =. MATHEMATICAL PRACTICES Reason Abstractl and Quantitativel 17. The blades of a propeller etend feet from the center of an engine. Carlos created two graphs depicting the vertical distance of points A and B above or below the center of the engine as the blades spin at a constant rate. In both graphs, Carlos let t = 0 represent the time the engine began spinning. Use the vocabular from this activit to describe an similarities and eplain an differences between the two graphs. A B 015 College Board. All rights reserved. 1 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Trigonometric Functions and the Unit Circle Wheels Revisited Lesson 1-1 The Unit Circle ACTIVITY 1 Learning Targets: Label points on the unit circle. Use the unit circle to find trigonometric values. SUGGESTED LEARNING STRATEGIES: Summarizing, Paraphrasing, Create Representations, Look for a Pattern, Quickwrite Suppose that a two foot diameter wheel has a paint mark at a point on the same horizontal line as the center of the wheel at a time t = 0 and that the spoke aligned with the paint spoke is denoted as spoke s. Suppose also that the wheel turns in the direction shown b the arrow in the figure below. 1. Let the origin represent the center of the biccle wheel and the position of spoke s at t = 0 represent the initial side of an angle θ in standard position. As the biccle wheel rotates, let spoke s represent the terminal side of angle θ. a. Some degree measures of the angle of rotation of spoke s are shown in the figure below. Use the smmetr of the figure to label the angle of rotation for each of the other positions shown for spoke s. 0 5 0 0 015 College Board. All rights reserved. b. Convert each of the degree measures in the figure above to radians and record the answers on the figure in part a. Paint spot when t = 0 Spoke s Activit 1 Trigonometric Functions and the Unit Circle 1

ACTIVITY 1 Lesson 1-1 The Unit Circle The -coordinate of the position of the paint spot is the horizontal distance to the right or left of the center of the wheel. The -coordinate is the vertical distance above or below the center of the wheel. Once again, label the rotation of spoke s in both degrees and radians. Then use our knowledge of special right triangles and the smmetr of the figure to label the coordinates of the paint spot for each of the rotations shown in the figure. MATH TERMS A circle with a radius of 1, centered at the origin, is known as a unit circle. MATH TIP The trigonometric functions cosine and sine are defined as follows: cos θ = r where + = r sin θ = r where + = r P(, ) r θ O The figure shown above is known as the unit circle. The unit circle is a circle of radius 1. It is used as a tool to recall trigonometric values of special angles. You should be able to reproduce the unit circle from memor quickl.. Recall the definition of sine of θ and eplain how ou can use the unit circle to find the value of sin θ for an angle on the unit circle. If P(, ) is an point on the terminal side of an angle θ in standard position, then cosine of θ is defined as cos θ = r where + = r.. Construct viable arguments. Eplain how ou can use the unit circle to find the value of cos θ for an angle on the unit circle.. Use the unit circle to give the eact value of each of the following. a. cos 5 b. sin π 015 College Board. All rights reserved. c. sin 180 d. cos 7 π 1 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1-1 The Unit Circle ACTIVITY 1 5. Once again, consider the paint spot on the biccle wheel and the rotation of spoke s. a. Give the eact value of the slope of spoke s for each angle given in the table below and the value of the slope correct to three decimal places. Then use our calculator to evaluate the tangent of the angle correct to three decimal places. Angle θ Eact slope Approimate slope tan θ π 15 π b. Epress regularit in repeated reasoning. Based on observations from the table in part a, write a definition for tangent of θ in terms of,, and r. MATH TIP When using a calculator to find trigonometric values, alwas choose the correct mode, either radians or degrees. CONNECT TO AP According to the AP Calculus course description, students must know the values of the trigonometric functions at the numbers 0, π, π, π, π, and their multiples. Being able to give these values quickl and accuratel in a wide variet of problem settings is a useful skill in AP Calculus. 015 College Board. All rights reserved.. Use the unit circle and the definition of tangent of θ to give the eact value of each of the following. a. tan 5 b. tan π c. tan 180 d. tan 7 π 7. Compare the values of tan 0 and tan 90. 8. Compare the values of tan 0 and tan 180. MATH TIP Recall that the slope of a line is defined as 1, where ( 1, 1 ) 1 and (, ) are an two points on the line. Activit 1 Trigonometric Functions and the Unit Circle 15

ACTIVITY 1 Lesson 1-1 The Unit Circle Do ou notice a pattern in the - and -coordinates of points on the unit circle? In this figure, each of the four points, P 1, P, P, and P, can be described b an angle of measure θ, using different initial sides and rotation in different directions. P (, ) P 1 (, ) O θ P (, ) P (, ) P 1 is a point on the terminal side of a counterclockwise rotation of an angle in standard position of θ degrees. P is a point on the terminal side of an angle in standard position (a counterclockwise rotation of measure 180 θ degrees from the positive -ais) or a clockwise rotation of θ degrees from the negative -ais. P is a point on the terminal side of an angle in standard position (a counterclockwise rotation of measure 180 + θ degrees from the positive -ais) or a counterclockwise rotation of θ degrees from the negative -ais. P is a point on the terminal side of a clockwise rotation of θ degrees or a counterclockwise rotation of θ degrees from the positive -ais. Eample A Given a 1 angle in standard position that intersects a unit circle with point P(0.978, 0.08) on the terminal side of a 1 angle, the points of intersection for the terminal sides of three other angles can be found. A 18 angle intersects the unit circle at P( 0.978, 0.08), because 18 = 180 1. A 19 angle intersects the unit circle at P( 0.978, 0.08), because 19 = 180 + 1. A 8 angle intersects the unit circle at P(0.978, 0.08), because 8 = 0 + ( 1). Using the definitions of sine, sin θ =, we see that sin 1 = sin 18, r sin 1 = sin 19, and sin 1 = sin 8. 015 College Board. All rights reserved. 1 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1-1 The Unit Circle ACTIVITY 1 Tr These A For these items, use a well-labeled drawing to justif our answer. a. Make use of structure. Given that P(0., 0.90) is the point where the terminal side of a 5 angle in standard position intersects a unit circle, find sin 5 and cos 5, correct to three decimal places. b. Given that cos π = +, find the eact values of cos 1 11 π and 1 cos. 1π 9. Given sin π 10 = 0.09, find between π and π such that sin = 0.09. MATH TIP Recall that 180 = π radians. 10. Given tan π =.078, find tan 8 π. 5 5 11. Given P( 0.85, 0.9) is the point where the terminal side of an n angle in standard position intersects a unit circle, find sin n, cos n, and tan n. Check Your Understanding 015 College Board. All rights reserved. 1. Epress regularit in repeated reasoning. Write an equation that relates tan θ and tan (180 θ). 1. Given that P(0.9, 0.8) is the point associated with a π 8 radian angle on the unit circle, what angle between 0 and π is represented b the point P( 0.9, 0.8)? Activit 1 Trigonometric Functions and the Unit Circle 17

ACTIVITY 1 Lesson 1-1 The Unit Circle LESSON 1-1 PRACTICE 1. Given cos 7 = 0., eplain how ou can use the Pthagorean Theorem to find the coordinates of the point representing 7 on the unit circle. 15. Use appropriate tools strategicall. Eplain how ou can use the trigonometr functions on our calculator to find the coordinates of the point representing 5 on the unit circle. 1. Given 0 < t < π, sin t = p, and cos t = q, complete the table. Let be an angle between 0 and π. π t π + t sin cos tan p q 17. Critique the reasoning of others. Dlan believes that for ever angle, 0 < < π, there is a second angle, 0 < < π, such that and cos = cos. Is Dlan correct? Eplain. Be sure to use correct mathematical terms to support our reasoning and that our sentences are complete and grammaticall correct. 015 College Board. All rights reserved. 18 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1- Reciprocal Trigonometric Functions ACTIVITY 1 Learning Targets: Define the reciprocal trigonometric functions using the unit circle. Evaluate all si trigonometric functions for an angle in standard position. SUGGESTED LEARNING STRATEGIES: Quickwrite, Think-Pair-Share, Identif a Subtask The reciprocal functions of sine, cosine, and tangent are cosecant, secant, and cotangent, respectivel. 1. Given that P(, ) is an point on the terminal side of an angle θ in standard position, and + = r. Define each of the si trigonometric functions in terms of,, and r. sine of θ : sin θ = cosecant of θ : csc θ = cosine of θ : cos θ = secant of θ: sec θ = tangent of θ : tan θ = cotangent of θ : cot θ =. Use the unit circle and the definitions of the reciprocal trigonometric functions to give the eact value of each of the following. a. sec 5 015 College Board. All rights reserved. b. cot π c. sec 70 d. csc 7 π Activit 1 Trigonometric Functions and the Unit Circle 19

ACTIVITY 1 Lesson 1- Reciprocal Trigonometric Functions So far we have onl considered trigonometric values for angles on the unit circle. Here are some other eamples for which the definitions of the trigonometric functions can be applied. Eample A Let (, ) be a point on the terminal side of θ, an angle in standard position. Find the values of sine, cosine, tangent, cosecant, secant, and cotangent of θ. We know that = and =. So r = + = + ( ) = 5 = 5 Therefore, appling the definitions from Item 1, we know that sin θ = 5 cos θ = 5 tan θ = cscθ = 5 secθ = 5 cot θ = Tr These A Given a point P on the terminal side of θ, an angle in standard position, find the eact values of sine, cosine, tangent, cosecant, secant, and cotangent of θ. a. P ( 5, 1) MATH TERMS Two circles with the same center are concentric circles. b. P (, ). Given P(, ) and P(, 8) are points on concentric circles. (, ) is a point on the terminal side of, an angle in standard position, and (, 8) is a point on the terminal side of, another angle in standard position. Both angles are between 0 and π. a. Which point is on the larger of the two circles? Eplain how ou know. b. Which angle has a greater measure? Eplain how ou know. 015 College Board. All rights reserved. O 0 SpringBoard Mathematics Precalculus, Unit Trigonometric Functions

Lesson 1- Reciprocal Trigonometric Functions ACTIVITY 1 Eample B Given that sin θ = and that cos θ < 0, find the values of the other five 5 trigonometric functions of θ. From the definition of sine, we know that = and r = 5. So r = + 5 = + = ± 1. Since cos θ < 0, we know that = 1. Therefore, appling the definitions from Item 1, we know that if sin θ = 5, then cscθ = 5 cos θ = 1 5 tan θ = = 1 1 secθ = 5 1 or 1 1 cot θ = 1 = or 1 5 1 1 Tr These B Find the values of the si trigonometric functions of θ, given the following information. a. tan θ = ; the terminal side of θ is in Quadrant III b. sec θ = 5 ; sin θ < 0 MATH TIP Recalling that r is alwas positive, ou can use the definitions of the trigonometric functions to determine the signs of the functions in each quadrant. 10 015 College Board. All rights reserved. 10 8 Quad II (, +) Sin θ: + Cos θ: Tan θ: Quad III (, ) Sin θ: Cos θ: Tan θ: + 8 8 10 Quad I (+, +) Sin θ: + Cos θ: + Tan θ: + Quad IV (+, )> Sin θ: Cos θ: + Tan θ: A function and its reciprocal function will have the same sign. 8 10 Activit 1 Trigonometric Functions and the Unit Circle 1