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Name:_____________________ Math 2412 Activity 4(Due by May 2) 1. Use properties of similar triangles to find the values of x and y. x 2y 7 74 x5 x y 14 74 21 2. For the angle in standard position with the point 5,12 on its terminal side, find the values of the six trigonometric functions: sin cos tan csc sec cot 5,12 3. Find one solution of the equation sin 2 10 cos 3 10 . {Hint: cos x sin 90 x .} 4. Find all the trigonometric function values of , if csc 2 and is in Quadrant III. sin cos tan csc 2 sec cot 5. Find the exact value of each labeled part: a m q 45 n a n 60 q 6. Find all the exact trigonometric function values of 1590 . sin1590 cos1590 tan1590 csc1590 m 7 sec1590 cot1590 7. Solve the right triangle to the nearest tenth of a degree and tenth of a foot: A 89.5 ft. b 47.9 m A B a C a b 8. Solve the right triangle to the nearest degree and the nearest foot: A c 137 ft. C 156 ft. B 9. Find h to the nearest tenth. h 35 21 x 135 x x 135 x 135 {Hint: cot 35 and cot 21 .} h h h h 10. Find h to the nearest tenth. 135 35 21 x {Hint: cot 35 h x xh x h and cot 21 .} 135 135 135 135 11. Find the area of the indicated sector: 5 8 12. Find the measure of the central angle, , in radians. 5 20 13. The rotation of the larger wheel causes the smaller wheel to rotate. Find the radius of the larger wheel if the smaller wheel rotates 90 when the larger wheel rotates 60 . 12 ft. 14. Graph the function y 2cos x on the interval 0,2 . r 15. Graph the function y 32 sin 32 x on the interval 0,3 . 3 4 3 9 4 3 2 16. Graph the function y 3sin 12 x 3 on the interval 0,4 . 2 3 4 5 17. Graph the function y 3cos 4 x 1 on the interval 4 , 4 . 4 8 8 4 1 18. Graph the function y sin 34 x 8 2 on the interval 2 6 5 6 3 2 6 , 176 . 13 6 17 6 19. Graph the function y 3sec 2 x 3 on the interval 6 5 12 6 , 76 . 7 6 11 12 2 3 20. Graph the function y 2csc x 2 1 on the interval 2 , 32 . 2 2 3 2 1 21. Graph the function y sec x 2 2 on the interval 2,4 . 2 x 22. Graph the function y tan on the interval 2 4 2 3 2 2 , 52 . 2 5 2 3 23. Graph the function y 3cot 4 x 2 on the interval 4 ,0 . 4 3 16 8 16 24. Graph the function y 4tan 3 x 2 on the interval 3,6 . 25. Determine the range of the following functions: a) y 3sin 2 x 7 b) y 2sec 2 x 11 8 26. Verify the identity cos4 x sin 4 x 2cos2 x 1 tan 2 x sec2 x sec x 27. Verify the identity . 2 2 cos x sin x sec x 2cos x 28. Show that the equation cos2 x cos x sin x is not an identity by demonstrating that for a specific value of x it is false. 29. Show that the equation sin 2 x sin x cos x 1 is not an identity by demonstrating that for a specific value of x it is false. 30. Find the exact value of cos 165 . 31. Find an exact value of that makes cot 10 tan 2 20 32. Verify the identity cos x 90 sin x sin 2 x 1 cos 2 x . 33. Find the exact value of cos 14 cos 29 sin 14 sin 29 . 34. Find the exact value of tan 512 tan 4 . 1 tan 512 tan 4 35. Find the exact value of sin165 . 36. Verify the identity tan x y tan y x 37. Verify the identity sin x y cot x cot y . cos x y 1 cot x cot y 38. Find the exact value of cos2 12 sin 2 12 . 39. Find the exact value of 4sin 22.5 cos 22.5 . 40. Verify the identity 2 tan x tan y . 1 tan x tan y 1 cos 2 x cot x . sin 2 x true. 1 tan 2 x 41. Verify the identity cos2 x . 1 tan 2 x 42. Find the exact value of cot 2 , if tan 5 and 90 180 . 2 43. Verify the identity tan 2x csc x cot x . 1 tan 2 2x 44. Verify the identity cos x . 1 tan 2 2x 2 45. Find the exact value of sin 1 . 2 2 46. Find the exact value of sec1 3 1 47. Find the exact value of tan 2cos 1 . 4 3 5 48. Find the exact value of cos sin 1 cos 1 . 5 13 1 49. Find the exact value of sin 2sin 1 . 3 50. Solve the equation cos2 cos 2 0 on the interval 0,2 . 51. Solve the equation 4sin 2 1 0 on the interval 0,2 . 52. Solve the equation sec2 tan 1 on the interval 0,2 . 53. Solve the equation cos2 sin 2 on the interval 0,2 . 54. Solve the equation cos 2x 0 on the interval 0,2 . 55. Solve the equation cos 2 x sin 2 x 0 on the interval 0,2 . 56. Solve the equation tan 2x sin x on the interval 0,2 . 57. Solve the equation sin x cos x 1 on the interval 0,2 . 2 Sketch the solutions of the following polar coordinate equations. 58. r 1 sin 59. r 1 2cos Find the points of intersection of the solution curves of the following pairs of polar coordinate equations. 60. r 1 cos , r cos 61. r 2cos3 , r 1 Find the points of intersection of the curves defined by the following parametric equations. x 1 t x 1 s ; 3 t 2 62. ; 3 s 2 and y 2 2s y t2 1 x sec s x 2cos t ;0 t 2 63. y 3sin t ; 3 s 3 and y tan s x cos s x cos t ;0 t 2 64. y sin 2t ;0 s 2 and y 12 sin s 65. Find the exact value of each part labeled with a variable. 8 30 x y w 60 z 66. The tires of a bicycle have a radius of 1.25 ft, and are turning at the rate of 5 revolutions per second. How fast is the bicycle traveling in feet per second? 67. If tan x .75 and cos x .8 , then find the value of tan x cos x . 68. Find the exact value of cos . 12 {Hint: 12 3 4 and cos A B cos A cos B cos A cos B .} 5 69. Find the exact value of tan . 12 {Hint: 70. Find the exact value of cos 5 tan A tan B .} and tan A B 12 6 4 1 tan A tan B 11 . 12 11 11 A 1 cos A 6 .} {Hint: cos and 12 2 2 2 Find the exact value of the following: 1 71. sin sin 1 12 4 72. sin 1 sin 3 2 73. cos sin 1 3 74. sin tan 1 2 1 75. tan cos 1 4 For each of the following, find sin x y , cos x y , tan x y , and the quadrant of x y . 76. sin x 1 4 , cos y , x in quadrant I, y in quadrant IV 10 5 2 1 77. sin y , cos x , x in quadrant II, y in quadrant III 3 5 Find the sine and cosine of the following 1 78. B , given cos 2 B , B in quadrant IV 8 5 79. 2y , given sec y , sin y 0 3 Find the following: 3 A 80. sin , given cos A , with 90 A 180 b) sin 2x , given sin x .6 , with 4 2 2 x 1 81. sin y , given cos 2 y , with y 3 2 Exactly solve the following trigonometric equations on the interval 0,2 . x x 85. csc sin 3 3 84. sec 4 2 x 4 82. sin 2 x 1 83. 3cos2 x 2cos x 1 0 86. sin x sin 2 x 87. cos 2 x cos x 0 x 90. cos 1 2 91. sin x 2 cos4 x 1 88. sin 2 x 2cos 2 x 89. 92. 6sin 2 x 17sin x 12 0 6 93. Sketch the graph of the solution to the polar coordinate equation r sin 2 . 1 r 4 1 2 3 4 2 sin3x 1 0 5 4 3 2 2 7 4 94. Sketch the graph of the solution to the polar coordinate equation r 1 cos . r 2 1 2 3 2 2 95. Find the points of intersection of the solution curves of the polar coordinate equations r 2 cos2 and r 2 sin . 96. Find the points of intersection of the solution curves of the polar coordinate equations r 2sin and r sin cos . 97. Graph the function y tan x 1 on the interval 2 , 2 . 98. Graph the function y sin 2 x on the interval 0, . 99. Determine the range of the function y 8sin 5x 7 . 98. If cos x 13 , then find the exact value of sin x tan x sin x cot x . Find the exact value of the following. 4 100. sin 2cos 1 5 {Hint: sin 2 A 2sin A cos A .} 1 2 101. sin sin 1 sin 1 4 3 {Hint: sin A B sin A cos B cos A sin B .} 1 102. tan 12 sin 1 3 {Hint: tan A sin A 1 cos A .} 2 1 cos A sin A 103. cos 12 sin 1 14 104. Sketch the graph of the solution to the polar coordinate equation r cos 2 . r 1 4 1 2 3 4 5 4 3 2 7 4 2 105. Sketch the graph of the solution to the polar coordinate equation r 1 2sin . r 3 1 2 7 6 3 2 11 6 2 1 106. Find the points of intersection of the solution curves of the polar coordinate equations r 1 sin and r 3sin . 107. Find the points of intersection of the solution curves of the polar coordinate equations r 2sin 2 and r 1. 108. Find the area of the region that is inside the solution curve of r 2sin but outside the solution curve of r sin . 109. Given that a 4i 3 j and b 2i j and another vector r 6i 7 j , find numbers k and m so that r ka mb . 110. Express c in terms of a and b , given that the tip of c bisects the line segment. b a 111. For what values of x are xi 11 j and 2xi xj orthogonal? c 112. Given that a i xj k and b 2i j yk , find all values of x and y so that a b and a b . 113. Use the dot-product to show that an angle inscribed in a semi-circle is a right angle. (Look at a b a b .) a b a b a b b 114. Show that the sum of the squares of the lengths of the diagonals of a parallelogram equals the sum of the squares of the lengths of the four sides. a 2 Expand a b a b 2 a b by using the dot-product. b b a b a 115. It looks as if a b and a b are orthogonal. Is this mere coincidence, or are there circumstances where we would expect the sum and difference of two vectors to be orthogonal? Find out by expanding a b a b 0 . a b b a a b b 116. Given vectors a and b , let m a and n b , show that a) na mb and na mb are orthogonal. b) c na mb bisects the angle between a and b . 117. Find all vectors v in the plane so that v 1 and v i 1 . Graph each parabola. 118. x 2 4 y 119. y 1 4 x 2 2 120. x2 8x 8 y Graph each ellipse. x2 y 2 121. 1 25 16 4 x 2 16 y 2 122. 1 81 25 123. 6 x 2 5 y 2 30 124. 9x2 18x 4 y 2 8 y 23 0 . Graph each hyperbola. x2 y 2 125. 1 16 25 126. 4 y 2 x 2 1 127. 4 x 2 25 y 2 100 128. 9 x2 18x 4 y 2 8 y 31 0 . 129. Find an equation for the parabola with focus of 4,4 and directrix of y 2 . 130. Find an equation of the hyperbola satisfying the given conditions: Endpoints of transverse axis: 4,0 , 4,0 ; asymptote y 2 x 131. Solve the system x2 y 2 9 x y 9 2 2 . Solve the following systems of equations. Check to see if your answer agrees with the graph. 132. x y 1 (line) y x 2 1 (parabola) 133. x 2 y 2 5 (circle) 3x y 5 (line) 134. 136. 4 x 2 y 2 4 (hyperbola) 4 x 2 y 2 4 (ellipse) y x 2 2 x 1 (parabola) y 1 x 2 (parabola) 135. 3x 2 4 y 2 16 (ellipse) 2 x2 3 y 2 5 137. (hyperbola) y x 2 2 (parabola) x 2 4 y 2 16 (ellipse) 138. Find the values of x and y in the figure. x 10 y 17 9