Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine

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1 14A Ready To Go On? Skills Intervention 14-1 Graphs of Sine and Cosine Find these vocabulary words in Lesson 14-1 and the Multilingual Glossary. Vocabulary periodic function cycle period amplitude frequency phase shift Identifying Periodic Functions Identify whether the function is periodic. If the function is periodic, give the period. A. Does the function repeat exactly at regular intervals? Is the function periodic? B. Does the function repeat exactly at regular intervals? Is the function periodic? Identify the period by using the start and finish of one cycle. The period of this function is. Stretching or Compressing Sine and Cosine Functions Using f (x) cos x as a guide, graph g (x) 3 cos 1 x. Identify the amplitude and period. Step 1 Identify the amplitude and period. Because a, the amplitude is a. Because b Step Graph. The curve is vertically, the period is b 1. by a factor of and horizontally by a factor of. The parent function f (x) has x-intercepts at n and g (x) has x-intercepts at. 5 Holt Algebra

2 14A Ready To Go On? Problem Solving Intervention 14-1 Graphs of Sine and Cosine You can use periodic functions to solve science problems. The torque applied to a bolt is given by (x ) Fr cos x, where r is the length of the wrench in meters, F is the applied force in newtons, and x is the angle between F and r in radians. Graph the torque for a 0.6 meter wrench and a force of 35 newtons for 0 x. What is the torque for an angle of 4? Understand the Problem 1. What two questions are you being asked?. What important information are you given in the problem? Make a Plan 3. To graph the function, find the and. 4. To find the torque, look at the graph when x. Solve 5. The amplitude of the function is 0.6( ). 6. The period of the function is. 7. Graph the function for 0 x. 8. What is the torque when x 4? Look Back 9. Substitute your solution into the original function. (x) Fr cos x 35 (0.6) cos x cos x x Is your answer approximately? 4 6 Holt Algebra

3 14A Ready To Go On? Skills Intervention 14- Graphs of Other Trigonometric Functions Transforming Tangent Functions Using f (x) tan x as a guide, graph g (x) tan 1 x. Identify the period, 4 x-intercepts, and asymptotes. Step 1 Identify the period. Because b, the period is. Step Identify the x-intercepts. The first x-intercept occurs at x. Because the period is, the x-intercepts occur at n, where n is an. Step 3 Identify the asymptotes. Because b, the asymptotes occur at x n or x n. Step 4 Graph using all of the information about the function. 7 Holt Algebra

4 14A Ready To Go On? Quiz 14-1 Graphs of Sine and Cosine Identify whether each function is periodic. If the function is periodic, give the period Using f (x) sin x or f (x) cos x as a guide, graph each function. Identify the amplitude and period. 5. d (x) 3 sin x 6. g (x) sin 1 x y 3π/ π π/ π/ π 3π/ x Amplitude: Period: Amplitude: Period: Using f (x) sin x or f (x) cos x as a guide, graph each function. Identify the x-intercepts and phase shift. 7. d (x) sin x 4 8. g (x) cos x 3 4 y 1 3π/ π π/ π/ π 3π/ 1 x x-intercepts: x-intercepts: Phase shift: Phase shift: 8 Holt Algebra

5 14A Ready To Go On? Quiz continued 9. The torque applied to a bolt is given by (x ) Fr cos x, where r is the length of the wrench in meters, F is the applied force in newtons, and x is the angle between F and r in radians. Graph the torque for a 0.4 meter wrench and a force of 350 newtons for 0 x. What is the torque for an angle of 3 8? 14- Graphs of Other Trigonometric Functions Using f (x) tan x as a guide, graph each function. Identify the period, x-intercepts and asymptotes. 10. d (x) tan(x ) 11. h (x) 1 tan (x) Period: x-intercepts: Asymptotes: Period: x-intercepts: Asymptotes: Using f (x) cot x as a guide, graph each function. Identify the period, x-intercepts and asymptotes. 1. d (x) cot x 13. g (x) cot x Period: x-intercepts: Asymptotes: Period: x-intercepts: Asymptotes: 9 Holt Algebra

6 14A Ready To Go On? Enrichment PERIODIC FUNCTIONS IN THE REAL WORLD A function is said to be periodic if the dependent variable takes on the same set of values repeatedly as the independent variable changes. For example, as you breathe, the volume of air in your lungs changes over time. The graph shown at right represents a reasonable graph for the situation. This function is periodic. In Exercises 1 3, sketch a reasonable graph for the situation. Then tell whether the function you graphed is periodic. 1. The depth of the water at a beach depends on the time of day due to the motion of the tides.. The distance required to stop a car depends on the speed of the car at the time of braking. Volume 0 Time 3. A gymnast jumps up and down on a trampoline. Her distance from the floor depends on time. Depth Distance Distance Time Speed Time The graph shown at right is periodic. It displays normal blood pressure ranges. The changes in pressure from systolic (from heart to arteries) to diastolic (from veins to heart) create the pulse. 4. Use the graph and determine the following: a. the domain b. the maximum and minimum values Pressure, in mm mercury p Systolic pressure Diastolic pressure Time, in seconds t c. the period y 5. a. On the axes at right, draw a graph of the step function shown below. f (x) { 1 if n 1 x n 0 if n x n 1 where n is an integer 0 x b. Is the function periodic? If so, state the period. 30 Holt Algebra

7 Ready To Go On? Skills Intervention 14-3 Fundamental Trigonometric Identities Proving Trigonometric Identities Prove each trigonometric identity. To prove that an equation is an identity, alter one side of the equation until it is the same as the other side. Justify your steps using the fundamental identities. A. csc sec cot csc sec cot csc 1 csc 1 Choose the right-hand side to modify. Use Reciprocal and Ratio Identities. csc B. co s (1 ta n ) 1 co s (1 ta n ) 1 Use the Reciprocal Identity. Choose the left-hand side to modify. co s 1 Use the Pythagorean Identity. co s 1 1 Use the Reciprocal Identity. co s 1 Multiply. 1 1 Using Trigonometric Identities to Rewrite Trigonometric Expressions Rewrite 1 si n in terms of a single trigonometric function. sin cos 1 si n sin cos sin cos sin Use the Pythagorean Identity. Use the Ratio Identity. 31 Holt Algebra

8 Find this vocabulary word in Lesson 14-4 and the Multilingual Glossary. Evaluating Expressions with Sum and Difference Identities Find the exact value of each expression. A. cos 75 Ready To Go On? Skills Intervention 14-4 Sum and Difference Identities Vocabulary rotation matrix Write 75 as the sum of 30 and 45 because the trigonometric values of 30 and 45 are known: cos 75 cos cos cos sin sin Apply identity for cos (A B). Evaluate. B. sin 1 Write 1 as the difference of and because the trigonometric values of and are known sin 1 sin sin cos cos sin Apply identity for sin (A B). Evaluate. 3 Holt Algebra

9 Ready To Go On? Skills Intervention 14-5 Double-Angle and Half-Angle Identities Evaluating Expressions with Double-Angle Identities Find sin if cos 3 and Step 1 Find sin to evaluate sin sin cos. Use the reference angle. In Quadrant II, cos 3 and y Use the Pythagorean Theorem. y Solve for y. y y sin Step Find sin. sin sin cos Find sin. Apply the identity for sin. Substitute known values. Using the Pythagorean Theorem with Half-Angle Identities Find cos if cos 3 and cos 1 cos 1 Apply the identity for cos. Substitute known values. Choose the positive square root since is in Quadrant I. 33 Holt Algebra

10 Ready To Go On? Skills Intervention 14-6 Solving Trigonometric Equations Solving Trigonometric Equations with Infinitely Many Solutions Find all solutions of 1 cos 0 where is in radians. Solve for over the principal values of cosine: cos 0 cos Subtract 1 from each side. cos Divide both sides by. co s 1 Apply the inverse cosine. or Find when cos 1. Use the period of cosine to find all solutions. The period of cosine is. n or n Solving Trigonometric Equations with Trigonometric Identities Solve cos sin 0 for cos sin 0 sin 0 Substitute 1 si n for cos. si n sin 1 0 Multiply both sides by 1. ( sin )(sin ) 0 Factor. sin 0 or sin 0 Apply the Zero Product Property. sin or sin Solve. si n 1 or si n 1 Apply the inverse sine. or, Find. 34 Holt Algebra

11 Ready To Go On? Problem Solving Intervention 14-6 Solving Trigonometric Equations You can solve trigonometric equations to make predictions about events in nature that are periodic. The average daily minimum temperature for Houston, Texas, can be modeled by T(x ) cos 6 (x 1) 76.85, where T is the temperature in degrees Fahrenheit, x is the time in months, and x 0 is January 1. On what dates is the temperature 80F? Understand the Problem 1. What is the question you must answer?. What important information is given to you about the variable x? Make a Plan 3. To answer the problem substitute for T (x ) and solve for. Solve 4. Solve the equation for x cos 6 (x 1) Subtract from both sides cos 6 (x 1) Divide both sides by cos 6 (x 1) Apply the inverse cosine. co s 1 ( ) 6 (x 1) Solve for x in Quadrants I and III. 5. Using an average of 30 days per month, convert your solutions to dates. Look Back 6. Check your answers by using a graphing calculator. Enter Y cos 6 (x 1) and Y 80. Graph the functions on the same viewing window and find the points of intersection of the curve and the line. Do the points of intersection match your solutions in Exercise 5? 35 Holt Algebra

12 Ready To Go On? Quiz 14-3 Fundamental Trigonometric Identities Prove each trigonometric identity. 1. cos csc () cot. co s tan sec sin 3. 1 ta n 1 co t ta n Rewrite each expression in terms of a single trigonometric function. 4. cot sin 5. 1 csc() 6. 1 co s se c Sum and Difference Identities Find the exact value of each expression. 7. sin cos 1 9. tan(75) Find each value if sin A 4 5 with 90 A 180 and if cos B 5 8 with 70 B sin(a B) 11. cos(a B) 1. sin(a B) 13. Find the coordinates, to the nearest hundredth, of the vertices of figure ABCD with A(1, 0), B(5, ), C(3, 4), and D(0, ) after a 150 rotation about the origin. 36 Holt Algebra

13 Ready To Go On? Quiz continued 14-5 Double-Angle and Half-Angle Identities 7 Find each expression if cos and sin 15. cos 16. tan 17. sin 18. cos 19. tan 0. Use half-angle identities to find the exact value of sin Solving Trigonometric Equations 1. Find all solutions of 1 cos 0 where is in radians. Solve each equation for cos 5 cos 3. 3 si n sin Use trigonometric identities to solve each equation for si n cos 1 5. cos 11 cos 6 6. The first sunrise in the United States each day is observed from Cadillac Mountain on Mount Desert Island in Maine. The time of the sunrise can be modeled by t (m) sin 6 (m 3) 5.485, where t is hours after midnight and m is the number of months after January 1. On what days does the sun rise at 6 A.M.? 5 A.M.? Use an average of 30 days per month. 37 Holt Algebra

14 Ready To Go On? Enrichment Solving Trigonometric Equations Involving Radicals Trigonometric equations containing radicals are solved in much the same way as equations without radicals, but you must clear the radical first to solve for the function of the angle. Then you can determine the measure of the angle. To solve cos 8 sin 1 for 0 360, follow these steps: cos 8 sin 1 ( cos ) ( 8 sin 1 ) Square both sides of the equation. 4 co s 8 sin 1 Use an identity to express the equation in terms 4(1 si n ) 8 sin 1 of a single trigonometric function. 4 4 si n 8 sin si n 8 sin 5 0 Solve for using the quadratic formula. 30 Solve each equation for tan 3 co t. csc x 3. 3 cos x To solve sin sin 0 for 0 360, follow these steps: sin sin 0 sin 1 cos 0 Use an identity to replace sin. si n 1 cos Isolate the radical and square both sides. 1 cos 1 cos Express as a single trigonometric function. 0, 40 Solve for. Solve each equation for cos sin 5. sin sin 38 Holt Algebra

5.1 Graphing Sine and Cosine Functions.notebook. Chapter 5: Trigonometric Functions and Graphs

Chapter 5: Trigonometric Functions and Graphs 1 Chapter 5 5.1 Graphing Sine and Cosine Functions Pages 222 237 Complete the following table using your calculator. Round answers to the nearest tenth. 2