π π π π Trigonometry Homework Booklet 1. Convert 5.3 radians to degrees. A B C D Determine the period of 15
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1 Trigonometry Homework Booklet 1. Convert 5.3 radians to degrees. A B C D Determine the period of y = 6cos x A. B. C. 15 D Determine the exact value of 5 tan A. B. C. D The point (m, n) is the intersection of the terminal arm of angle θ in standard position and the unit circle x + y = 1. Which expression represents? m A. m B. n C. D. n n m 5. Which of the following is an asymptote of y = sec x? A. x= 0 B. x= C. x= D. x= 4 6. Simplify: sin θ A. B. C. D. 7. Which expression is equivalent to cos x + cot x? sin x + 1 A. sec x B. csc x C. cot x D. tan x 8. Which expression is equivalent to sin x+ + sin x 3 3? 3 A. sin x B. sin x C. 3 sin x D. sin x 4 9. Solve: sin 3x+ tan x= 3, 0 x A. 1.31, 4.34 B..44, 3.85 C. 1.31, 1.57, 4.34, 4.71 D. 0,.44, 3.14, 3.85
2 10. A Ferris wheel has a radius of 18 metres and a centre C which is 0 m above the ground. It rotates once every 3 seconds. A platform allows a passenger to get on the Ferris wheel at a point P which is 0 m above the ground. If the ride begins at point P, when the time is t = 0 seconds, determine a sine function that gives the passenger s height, h, in metres, above the ground as a function of t. A. ht () = 18sin t+ 0 B. ht () = 18sin t C. ht () = 0sin t+ 18 D. ht () = 0sin t Determine the period of y = tan x. A. 1 radian B. radians C. radians D. radians 1. Given a circle with radius 10 cm, calculate the length of arc a which contains a sector angle θ = radians. A.5 cm B.10 cm C.10cm D.0cm Find the exact value of tan A. 3 B. C. D
3 14. Solve: cos x= x, 0 x< A B C D. no solution 15. The expression cos3x cos x sin 3xsin x is equal to A. sin x B. sin 5x C. cos x D. cos5x 16. Solve: cos 1= 0, 0 < x x A., B., C.,,, D.,,, Determine the maximum value of the function f ( x) = acos x+ d, where a > 0, d > 0. A. a B. d a C. a+ d D. a+ d 18. Simplify: 1 + cotθ cscθ A.cscθ B. C.cotθ D The terminal arm of angleθ in standard position passes through the point (m, n) sin + θ. where m > 0, n > 0. Determine the value of ( ) n m n m A. B. C. D. m + n m + n m + n m + n 0. A wheel of radius 30 cm has its centre 36 cm above the ground. It rotates once every 1 seconds. Determine the equation for the height, h, above the ground of a point on the wheel at time t seconds if this point has a minimum height at t = 0 seconds. A. ht () = 30 cos t+ 6 B. ht () = 30 cos t C. ht () = 30 cos t+ 36 D. ht () = 30 cos t Convert 10 to radians. A B..69 C D Determine an expression equivalent to secθ cotθ. A. 1 B. cotθ C. cscθ D. tanθ
4 3. Determine the exact value of 7 sec A. B. C. D. 4. Determine the period of the function y = 3cos 4x. A. B. C. 6 D Determine the range of the function y = sin3x+ 4. A. 6 y B. y C. 0 y 4 D. y 6 6. Solve: cosx+ 3 = 0, 0 x< A., B., C., D., Solve: sin x+ cos 3x= 1.5, 0 x<. A. 3.84, 4.37 B. 4.97, 5.1 C. 5.07, 5.58 D. 1.0, 1.90, 3.76, Simplify: sin ( x + ) A. sin x B. cos x C. sin x D. cos x 9. The two smallest positive solutions of sin 3x = 0.4 are x = 0.14 and x = Determine the general solution of sin 3x = 0.4. ( ) ( ) A. x= n, x= n, n is an integer B. x= n, x= n, n is an integer C. n n x= , x= , ( n is an integer) 3 3 D. n n x= , x= , 3 3 n is an integer 30. The function ht () ( t ) ( ) = 3.9sin gives the depth of water, h, metres, at any time, t hours, during a certain day. A cruise ship needs at least 8 metres of water to dock safely. Use the graph of the function to estimate the number of hours in the 4 hour interval starting at t = 0 during which the cruise ship can dock safely. A B C D. 9.36
5 31. Determine the amplitude of y ( x ) A. 5 B. 3 C. 4 D. 5 = 5sin Convert 135 to radians. A B. 1.9 C..36 D Determine the period of y = tan 4x. A. B. C. D Determine the exact value of 11 sec 6. A. B. C. D. 3 3 csc x Simplify: csc x A. cos x B. sin x C. cos x D. sin x 36. Solve: sinx+ 1= 0, 0 x< A., B., C., D., Solve: 3cos x= x, 0 x< A B. 0.5, 1.57 C. 0.67, 3.07 D. 0.95, Which equation represents the sine function graphed below? A. 4 y = 4sin x B. 4 y = 4sin x C. 3 y = 4sin x D. 3 y = 4sin x
6 39. A wheel rolling along the ground has a radius of 3 cm and rotates once every 8 seconds. At time t = 0, a point P on the outside edge of the wheel is touching the ground. Determine a cosine function that gives the height, h, of point P above the ground at any time, t, where h is in cm and t is in seconds. A. ht () = 3cos t 4 B. ht () = 3cos t C. ht () = 3 cos t+ 3 4 D. ht () = 3 cos t Determine the number of solutions for ( asin x+ a) ( bcos x c) = 0 for 0 x <, if 1 < a< b< c. A. 1 B. C. 3 D Convert 5 radians to degrees A. 90 B. 180 C. 70 D Determine the range of the function y= 4cosx. A. 4 y 4 B. y 6 C. 6 y D. y Solve: sin x cos x= 1, 0 x< A. 0, 5.07 B. 3.14, 4.3 C. 3.14, 4.36 D. 0.4, 1.89,.95, Determine the exact value of 5 cot A. 3 B. 3 C. D Determine the period of the function ( ) A. B. C. 4 D Solve: sin x+ 1= 0, 0 x< 1 x f x = sin A., B., C., D.,
7 tan θcsc θ 47. Determine an expression equivalent to. sec θ 3 A. tanθ B. cotθ C. tan θ D. tan θ 48. Simplify: cos( x) A. cos x B. sin x C. cos x D. sin x 49. A wheel with radius 0 cm has its centre 30 cm above the ground. It rotates once every 15 seconds. Determine an equation for the height, h, above the ground of a point on the wheel at time t seconds if this point has a maximum height at t = seconds. A. h= 0 cos ( t+ ) + 30 B. h= 0 cos ( t ) C. h= 30cos ( t+ ) + 0 D. h= 30cos ( t ) Determine a cosine equation that has the following general solution: 11 + n, + n, + n 6 6 ( ) ( ) ( ) ( ) A. cos x cos x+ = 0 B. cos x cos x+ 3 = 0 C. cos x cos x = 0 D. cos x cos x 3 = Give the exact value of 11 cos A. B. C. D. 5. Simplify: sin θ. A. 1 B. C. cscθ D. secθ An arc of length 5 cm subtends an angle of 30 at the centre of a circle with radius r. Determine the value of r. A B C D Determine the period of y = tan x. A. 1 B. C. D.
8 55. Solve: 3sin x= x+ 1, 0 x< A. 0.5 B. 1.87,.87 C. 0.54, 1.54 D. 0.54, Simplify: 3 sin + x A. sin x B. cos x C. sin x D. cos x 57. Solve: sin x= sin xcos x, 0 x< A. x= 0, B. x=, C. x= 0,,, D. x= 0,,, The terminal arm of angle θ in standard position passes through the point (, 5). Determine the value of secθ A. B. C. D Determine the range of the function y= bcos ax b, where a> 0, b> 0. A. b y 3b B. 3b y b C. b a y b+ a D. b a y b+ a 60. Determine the general solution for: 1 sin x = A n, + n (n is any integer) 1 1 B n, + n (n is any integer) C. + n, + n (n is any integer) D. + n, + n (n is any integer) Convert 150 to radians A. B. C. D Solve tan x+ sin x= 1 ( 0 x< ) A. 0.49, 4. B..06, 5.80 C. 0.49, 1.57, 4., 4.71 D. 1.57,.06, 4.71, 5.80
9 63. The diagram below shows the unit circle, determine A. p B. q C. p D. q 64. Determine the period of the function y = tan x 5 A. 5 B. 10 C. 5 D Which expression is equivalent to cos8 x + 1 A.cos 4 x B.sin 4 x C.cos 16 x D.sin 16x 66.
10 67. Solve: 3+ sinx= 0 ( 0 x< ) A., B., C., D. 7, y = asin x c + d where a, c and d are positive constants, determine the range of 68. In the function ( ) the new function formed if a is doubled a a Ad. y d+ Bd. a y d+ a a a C. d y d + D. d a y d + a 69. Determine the general solution to 3sin5x = 1 n n n n Ax. = , x= Bx. = , x= Cx. = n, x= n Dx. = n, x= n 70.
11 Written Questions: 1. Solve cos x+ cosx 1= 0 algebraically over the set of real numbers. (Give the general solution using exact values). Prove: = 1+ cosx tanx sin x sec x 1 3. Prove the identity: x( x x) sin tan + cot = 4. Prove: x + x x = sin x 1 cos x cos cos sin 5. Solve the following equation algebraically x+ x = x 3cos cos 0, 0
12 6. Prove the identity: ( ) sin θ cscθ sin θ tan θ= 7. A Ferris wheel has a radius of 5 m and its centre is 7 m above the ground. It rotates once every 40 seconds. Sandy gets on the Ferris wheel at its lowest point and then the wheel begins to rotate. a) Determine a sinusoidal equation that gives Sandy s height, h, above the ground as a function of the elapsed time, t, where h is in metres and t is in seconds. b) Determine the first time, t (in seconds), when Sandy will be 35 m above the ground. 8. Prove the identity: sin x sin x = tan 1 sin x 1+ sin x x 9. Prove the identity: cotθ secθ cscθ =
13 Key 1. D. C 3. B 4. B 5. C 6. D 7. C 8. B 9. A 10. A 11. C 1. D 13. A 14. A 15. D 16. C 17. C 18. B 19. A 0. D 1. D. A 3. D 4. A 5. D 6. A 7. A 8. C 9. D 30. D 31. D 3. C 33. A 34. D 35. A 36. D 37. D 38. D 39. C 40. A 41. D 4. C 43. C 44. D 45. D 46. C 47. B 48. A 49. B 50. D 51. D 5. D 53. C 54. A 55. D 56. D 57. D 58. C 59. B 60. B 61. C 6. A 63. A 64. A 65. A 66. A 67. B 68. B 69. A 70. B 1.. cos x+ cosx 1= 0 LHS RHS cos x+ cos x = 0 sinxcosx sec x 1 = ( cos x 1)( cos x+ ) = 0 1+ cosx tanx sinxcosx tan x 1 = cos x cos x+ = 0 1+ cos x 1 tanx sinxcosx ( cos x 1)( cos x+ 1) = 0 = tan x cos x 1 cos x = cos x = 1 sin x = cos x 5 x = + n, x = + n, x = + n tan x LHS = RHS sin x tan x+ cot x ( ) sin x cos x sinxcosx + cos x sin x sin x cos x sin xcosx + cos xsin x sin xcos x sin x+ cos x sin xcosx sin xcos x 1 sin xcosx sin xcos x
14 4. cosx + cos x sin x cosx + cos x sinxcosx cos x 1 cos ( + x) sinxcosx 1+ cos x 1 cos x sin x 1 cos x 1 cos x sin x 1 cos ( x) sin x sin x 1 cos sin x 1 cosx ( x) 5. 3cos x+ cos x = 0 cos + cos 6 = 0 x ( x )( x ) cos cos + 3 = 0 3 cos x cos x+ = cos cos + 1 = 0 ( x )( x ) x cos x= cos x= 1 3 x= 0.841,5.44 x= 6. LHS RHS sin θ cscθ tanθ 1 tanθ ( ) 1 sin θ tanθ 1 sin θ tanθ cos θ tanθ cos θ LHS = RHS 7. 10a) h= 5cos t b) 1.07 seconds
15 8. sin x sin x 1 sinx 1+ sinx sin x 1+ sin x sin x 1 sin x 1 sin x 1+ sin x sinx+ sin x sinx+ sin x 1 sinx 1+ sinx ( ) ( ) ( )( ) sin x 1 sin x sin x cos x tan x 9. cotθ cscθ 1 sin θ 1 sin θ 1 cos θ 1 secθ
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