Name Date Class. Identify whether each function is periodic. If the function is periodic, give the period
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1 Name Date Class 14-1 Practice A Graphs of Sine and Cosine Identify whether each function is periodic. If the function is periodic, give the period. 1.. Use f(x) = sinx or g(x) = cosx as a guide. Identify the amplitude and period. Then graph each function. 3. h(x) = 3sin4x 4. p(x) = cos(0.5x) Use f(x) = cosx as a guide. Graph the function. 5. g(x) = cos(x π) a. Identify the amplitude and period. b. Identify the phase shift. c. Identify the x-intercepts. d. Identify the maximum and minimum values. e. Use the information to graph the function Holt Algebra
2 Name Date Class 14-1 Practice B Graphs of Sine and Cosine Using f (x) = sinx or g(x) = cosx as a guide, graph each function. Identify the amplitude and period. 1. b(x) = 5sinπ x. k(x) = 3cosπ x Using f (x) = sinx or g(x) = cosx as a guide, graph each function. Identify the x-intercepts and phase shift. π 3. hx ( ) = sinx+ 4 π 4. hx ( ) = cosx 4 5. a. Use a sine function to graph a sound wave with a period of 0.00 second and an amplitude of centimeters. b. Find the frequency in hertz for this sound wave Holt Algebra
3 Name Date Class 14-1 Practice C Graphs of Sine and Cosine Using f (x) = sinx or f (x) = cosx as a guide, graph each function. Identify the amplitude, period, x-intercepts, and phase shift π hx ( ) = cos( π x). qx ( ) = sin x π 3. c(x) = 3cos(x + π) 4. hx ( ) = sinx 5. A manual metronome is an inverted pendulum that helps musicians play to the beat. The number of centimeters, C, that the tip of the pendulum is from a tabletop can be modeled by C(t) = cos 4π t + 1, where t is the time in seconds. a. Graph the height of the pendulum tip for periods. b. How high is the pendulum when 1 t = second? Holt Algebra
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5 Name Date Class 14- Practice A Graphs of Other Trigonometric Functions Graph each function. Use f (x) = tanx as a guide. Identify the period, x-intercepts, and asymptotes. 1. p(x) = tan(3x) a. Identify the period. b. Identify the x-intercepts. c. Identify the asymptotes. d. Use the information to graph the function. x. qx ( ) = tan 3. k(x) = tan(π x) Graph each function. Use f (x) = cotx as a guide. Identify the period, x-intercepts, and asymptotes. x 4. hx ( ) = cot a. Identify the period. b. Identify the x-intercepts. c. Identify the asymptotes. d. Use the information to graph the function bx ( ) = cot( x) 6. bx ( ) = cot( x) Holt Algebra
6 Name Date Class 14- Practice B Graphs of Other Trigonometric Functions Using f (x) = tanx and f (x) = cotx as a guide, graph each function. Identify the period, x-intercepts, and asymptotes. π x 3 1. gx ( ) = tan. tx ( ) = cot( x) 4 Using f (x) = cosx or f (x) = sinx as a guide, graph each function. Identify the period and asymptotes. x 1 3. kx ( ) = sec 4. qx ( ) = csc( x) 5. The rotating light on a lighthouse is 400 feet from a cliff and completes one full rotation every 10 seconds. The equation representing the distance, a, in feet that the center of the πt circle of light is from the lighthouse is at () = 400sec 5. a. What is the period of a(t)? b. Find the value of the function at t = Holt Algebra
7 Name Date Class 14- Practice C Graphs of Other Trigonometric Functions Using f (x) = tanx or f (x) = cotx as a guide, graph each function. Identify the period, x-intercepts, and asymptotes kx ( ) = tan( π x). 4 x cx ( ) = cot 3 Using f (x) = cosx or f (x) = sinx as a guide, graph each function. Identify the period and asymptotes. 3. ( ) sec 4 x kx = 4. q(x) = csc(x) 5. A strobe light is located in the center of a square dance room. The rotating light is 40 feet from each of the 4 walls and completes one full rotation every 6 seconds. The equation representing the distance, d, in feet that πt the center of the circle of light is from the light source is dt () = 40sec 3. a. What is the period of d(t)? b. Find the value of the function at t = Holt Algebra
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9 Name Date Class 14-3 Practice A Fundamental Trigonometric Identities Prove each trigonometric identity. sinθ 1. csc θ 1 cos θ = a. Modify the left-hand side. Replace 1 with a known identify. b. Simplify the denominator. c. Keep simplifying and substituting identities until the left side matches the right side.. cos 4 θ + cos θ sin θ + sin θ = 1 3. cscθ cotθ = secθ Rewrite each expression in terms of cosθ. Then simplify. cotθ 4. cscθ a. Rewrite each function in terms of sinθ or cosθ. b. Rewrite the fraction in simplest form. 5. sinθ cotθ 6. sin cos θ cos θ θ 7. sin θ + cos θ sec θ tan θ 8. Use the equation mg sinθ = µmg cosθ to determine the angle at which a desk can be tilted before a paperback book on the desk begins to slide. Assume µ = Holt Algebra
10 Name Date Class 14-3 Practice B Fundamental Trigonometric Identities Prove each trigonometric identity. 1. sin θ + sin θ cot θ = 1. cot θ cos θ = cot θ cos θ 3. tan θ tan θ sin θ = sin θ 4. sin θ + cos θ = sec θ + csc θ sin θ cos θ Rewrite each expression in terms of cosθ. Then simplify. 1+ cotθ 5. sinθ cosθ cotθ 6. cot θ (sin θ + cos θ ) 7. cos 4 θ sin 4 θ + sin θ 8. Use the equation mg sinθ = µmg cosθ to determine the angle at which a waxed wood block on an inclined plane of wet snow begins to slide. Assume µ = Holt Algebra
11 Name Date Class 14-3 Practice C Fundamental Trigonometric Identities Prove each trigonometric identity. cos θ 1 sin θ 1. = 1+ sinθ cosθ. sec θ + tan θ tan θ sec θ = cos θ + cot θ 3. tan θ sin θ = tan θ sin θ cot 4 θ = csc 4 θ cot θ Rewrite each expression in terms of sinθ and cosθ. Then simplify. 5. cot θ 6. cot θ cosθ cot θ 7. tan θ + 4 sec θ + 1 sin θ 8. Alan is using the equation mg sinθ = µmg cosθ to determine the coefficient of friction, µ, between a flat rock and a metal ramp. Find µ to the nearest hundredth if the rock begins to slide at Holt Algebra
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13 Name Date Class 14-4 Practice A Sum and Difference Identities Find the exact value of each expression. 1. cos10 a. Write 10 as a sum or difference of two known trigonometric values. b. Substitute the values into the correct sum or difference identity. c. Evaluate and simplify.. cos sin tan π tan π sin 3 7. π cos Let sin A = with 90 A 180 and tan B = with B 360. Find each value. 8. tan(a + B) 1 a. Use sin A = to find tana. 13 b. Substitute known values into the angle sum tan A+ tan B identity tan ( A + B) =. 1 tan AtanB c. Simplify. 9. tan(a B) 10. sin(a + B) 11. cos(a B) 1. The displacement, y, of a mass attached to a spring is modeled by π π yt () = 9.6cos () t 6, where t is the time in seconds. a. What are the amplitude and period of the function? b. Write the displacement using only the sine function. c. What is the displacement of mass when t = 6? 14-7 Holt Algebra
14 Name Date Class 14-4 Practice B Sum and Difference Identities Find the exact value of each expression. 1. cos10. sin tan55 4. tan 7 π 6 π 5. sin 1 3π 6. cos 4 Prove each identity. 3π 7. sinx = cos x π 8. cosx = sin x 1 8 Find each value if cos A = with 0 A 90 and if sin B = with B sin(a + B) 10. cos(a + B) 11. tan(a + B) 1. sin(a B) 13. cos(a B) 14. tan(a B) 15. Find the coordinates, to the nearest hundredth, of the vertices of triangle ABC with A(1, 0), B(10, 0), and C(, 6) after a 60 rotation about the origin. a. Write the matrices for the rotation and for the points. b. Find the matrix product. c. Write the coordinates. 16. A hill rises from the horizontal at a 15 angle. The road leading straight up the hill is 800 meters long. How much higher is the top of the hill than the base of the hill? 14-8 Holt Algebra
15 Name Date Class 14-4 Practice C Sum and Difference Identities Find the exact value of each expression. 1. cos300. sin( 15 ) 3. tan π tan π sin π cos 1 Prove each identity. 7. tan(π x) = tanx π 8. sin x = cos x 1 Find each value if sin A = with 90 A 70 and if 13 with 90 B sin B = 5 9. sin(a + B) 10. cos(a + B) 11. tan(a + B) 1. sin(a B) 13. cos(a B) 14. tan(a B) 15. Find the coordinates, to the nearest hundredth, of the vertices of triangle PQR with P(0, 1), Q(0, 4), and R(, 5) after a 45 rotation about the origin. 16. Find the coordinates, to the nearest hundredth, of the vertices of triangle FGH with F(0, 10), G(10, 0), and H(1, 1) after a 75 rotation about the origin. 17. Two pilots take off from the same airport. Mason heads due south. Nancy heads 3 west of south. After 400 land miles, how far is Nancy from Mason s route? 14-9 Holt Algebra
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17 Name Date Class 14-5 Practice A Double-Angle and Half-Angle Identities Find sin θ, cos θ, and tan θ for each set of conditions sin θ = for π < θ < 5 3π a. Find cosθ by substituting into formulas for sin θ and cos θ. b. Use the double-angle identity sin θ = sin θ cos θ to evaluate sin θ. c. Use the double-angle identity cos θ = cos θ sin θ to evaluate cos θ. d. Find tanθ for substituting into formula for tan θ. tanθ e. Use tan θ = to evaluate tan θ. 1 tan θ π cos θ = for < θ < π sin θ = for 3 π < θ < π sin θ = for 0 < θ < 8 π 1 tan θ = for π < θ < 3 3π Prove each identity. 6. sin θ + 1 = (sinθ + cosθ ) a. Expand the right side of the equation. b. Use the Associative Property of Addition. c. Use the double-angle identity for sinθ. d. Use the Pythagorean Identity. 7. sin θ tanθ cos θ = 8. cos θ 1 = cos θ Holt Algebra
18 Name Date Class 14-5 Practice B Double-Angle and Half-Angle Identities Find sin θ, cos θ, and tan θ for each π cos θ = for π < θ < sin θ = for 0 < θ < 10 π 3. sin θ = for 3 π < θ < π π tan θ = for < θ < π 6 Prove each identity. 5. cos θ = cos θ cosθ tan θ = sin θ Use half-angle identities to find the exact value of each trigonometric expression. 7π 11π 7. tan.5 8. cos 9. sin 1 1 Find sin θ, cos θ, and tan θ for each cos θ = and 70 < θ < sin θ = and 180 < θ < A water-park slide covers 100 feet of horizontal space and is 36 feet high. a. Write a trigonometric relation in terms of θ, the angle that the slide makes with the water surface. b. A new replacement slide will create an angle with the water surface that measures twice that of the original slide. The new slide will use the same horizontal space as the old slide. Write an expression that can be evaluated to find the height of the new slide. c. What is the height of the new slide to the nearest foot? Holt Algebra
19 Name Date Class 14-5 Practice C Double-Angle and Half-Angle Identities Find sin θ, cos θ, and tan θ for each cos θ = for 3 π < θ < π tan θ = for π < θ < 7 3π 3. 5 π cos θ = for < θ < π sin θ = for 0 < θ < 16 π Prove each identity. sin θ tanθ 5. cos θ = 6. sin 4θ = 4cosθ sinθ 8cosθ sin 3 θ tan θ Use half-angle identities to find the exact value of each trigonometric expression. 7. sin π cos tan 8 π Find sin θ, cos θ, and tan θ for each sin θ = for 90 < θ < cos θ = for 70 < θ < tanθ = 7 for 180 < θ < tan θ = for 70 < θ < Amanda wants to shorten the wheelchair ramp at her front door. The current ramp is 13 feet along the horizontal and the step is 9 inches high. If she doubles the measure of the angle that the ramp makes with the walkway, how long will the new ramp be to the nearest inch? Holt Algebra
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21 Name Date Class 14-6 Practice A Solving Trigonometric Equations Find all of the solutions of each equation. 1. 3sinθ = sinθ 3 a. Subtract sinθ from both sides and combine like terms. b. Solve for sinθ. c. Apply the inverse sine.. 5cosθ = 3cosθ tanθ = 9tanθ + 1 Solve each equation for the given domain. 4. cos θ cosθ = 1 for 0 θ 360 a. Add 1 to both sides. b. Factor or use the quadratic formula to solve for cosθ. c. Apply the inverse cosine. 5. tan θ = 9 for 0 θ sin θ = 3sinθ for 0 θ 360 Solve each equation for the given domain. Use trigonometric identities. 7. cos θ = sin θ + sinθ for 0 θ 360 a. Rewrite the equation in terms of sinθ. b. Solve for sinθ. c. Apply the inverse sine. 8. cos θ + sinθ = 1 for 0 θ sin θ tanθ = 1 for 0 θ The horizontal range, R, in feet that a baseball travels v sin θ is modeled by R( θ ) =. The initial velocity, v, 3 is 90 feet per second. The ball is pitched at an angle of θ degrees with the horizontal. At what angle(s) will the ball travel 00 feet? Holt Algebra
22 Name Date Class 14-6 Practice B Solving Trigonometric Equations Find all of the solutions of each equation. 1. 4tanθ = 5tanθ + 3. sinθ = cosθ 1 = 9cosθ 4. tanθ + 3 = 5tanθ Solve each equation for the given domain. 5. tan θ tanθ = 1 for 0 θ sin θ = 1 for 0 θ cos θ = 3cosθ for 90 θ cos θ cosθ = 3 for π θ π 9. 3tan θ = tanθ + 1 for 0 θ tan θ = tanθ + 6 for 180 θ 360 Use trigonometric identities to solve each equation for the given domain. 11. sinθ + sin θ = 0 for 0 θ cos θ = sin θ for 0 θ 360 cos θ sin θ = for 0 θ cos θ = sin θ for 0 θ The height of the water at a pier on a certain day can be modeled by π ht () = 4.8sin ( t+ 3.5) + 9, where h is the height in feet and t is the 6 time in hours after midnight. When is the height of the water 6 feet? Holt Algebra
23 Name Date Class 14-6 Practice C Solving Trigonometric Equations Find all of the solutions of each equation. 1. 4tanθ + 3 = tanθ. cosθ = 1 + 3cosθ Solve each equation for the given domain. 3. 4cos θ = 1 for 0 θ cos θ + cosθ 1 = 0 for 0 θ tan θ + 3 = 3tanθ + 3 tanθ for π θ π Use trigonometric identities to solve each equation for the given domain. θ 6. sin θ = cos θ for 0 θ cos = sin θ for 0 θ tan θ = cotθ for 0 θ θ cos = cos θ for 0 θ sin π π θ = 1 sin + θ 4 4 for 0 θ π 11. cos θ tanθ = sinθ for 0 θ cosθ + sinθ = 1 for 0 θ cos 3 θ sin 3 θ = 1 for 0 θ The number of hours of daylight, d, at Quintilla Falls can π be modeled by hd ( ) = 3.sin ( d 88) + 1.3, 18.5 where d is the number of days after January 1. On what dates are there 1 hours of sunlight? Assume the year is not a leap year Holt Algebra
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