Midterm Review January 2018 Honors Precalculus/Trigonometry Use the triangle below to find the exact value of each of the trigonometric functions in questions 1 6. Make sure your answers are completely simplified/rationalized. 1. sin θ 4. csc θ 2. cos θ 5. sec θ 3. tan θ 6. cot θ 7. Find the missing angle: 8. Find the exact value of a and b in the triangle. Your final answers should not be in decimal form. 9. Find the exact value of the hypotenuse in the triangle. Your final answer should not be in decimal form. Then give the value of θ in the triangle. Hypotenuse = θ = 10. Find the exact value of m and n in the triangle. Your final answers should not be in decimal form
11. Find the missing side(x) in the triangle below. Round your answer to the nearest tenth if necessary. 12. Find the length of the arc in the circle. Round to the nearest tenth if necessary. 13. Convert 675 to radians. Leave your answer in terms of π. 14. Convert 456 7 to degrees. 15. Give an example of one positive and one negative angle that is co-terminal with 110. 16. Let ( 2, 5) be a point on the terminal side of angle θ in standard position. Find the exact value of the 6 trigonometric functions of θ. 17. Let (0, -11) be a point on the terminal side of angle θ in standard position. Find the exact value of the 6 trigonometric functions of θ. 18. In what quadrant does θ lie if sec θ < 0 and tan θ > 0? 19. Suppose sec θ = @ and θ lies in Quadrant II, find the exact value of the 5 remaining trigonometric 7 functions of θ. 20. Suppose tan θ = 5 and 180 < θ < 270. Find the exact value of the 5 remaining trigonometric functions of θ. 21. Find the reference angle of: a) 280 b) 45 c) 130 d) 350 22. What is the reference angle of 5E6 E? 23. Why are reference angles useful?
Find the exact value of each of the following. You must have the unit circle memorized for the exam. 24. tan 750 25. cot π 26. sec ( 900 ) 27. tan 225 28. cos ( 210 ) 29. cos 556 7 30. csc ( 450 ) 31. cot ( 300 ) 32. csc 720 33. cot 5E6 G 34. cos 576 E Graph two periods of each function. Label the max and min on the x and y axis. Label the y- intercept if necessary. 35. y = 5 cos x 2 4 36. Graph: y = 5 4 sin x 6 E 5 7 37. Graph y = cos 4x + 6 4 38. Graph y = 2 sin 5 4 x + 6 7 x + 3
For # 39-44, label each of the given graphs with the correct equation (Sine, Cosine, Tangent, Secant, Cosecant or Cotangent). 39. 40. 41. 42. 43. 44. 45. What is the period of the sine function? 46. What is the period of the cosine function? 47. What is the period of the tangent function? 48. If the frequency of a function is 5, what is the period of the function? 49. What is the definition of amplitude? 50. What is the domain of the tangent function? @6
51. What is the range of the sine and cosine functions? 52. Which trigonometric functions are even? What does that tell you about their graphs? 53. Which trigonometric functions are odd? What does that tell you about their graphs? 60. If sec θ = L, and θ is in quadrant IV, find: 7 a) cos θ b) cos( θ) c) sec θ d) sin θ e) sin( θ) f) tan θ g) tan( θ) h) cot θ i) cot( θ) 61. If cos θ = 5 and tan θ > 0, find: 7 a) sec θ b) sec( θ) c) cot(θ) d) csc θ e) sin( θ) 62. If sin θ = 0.87, what is cos 6 4 θ? What is cos θ 6 4? 63. If cot θ = 1.43, what is tan 6 4 θ? What is tan θ 6 4? 64. If sin θ < 0 and tan θ < 0, in what quadrant does θ lie? 65. Two angles that have the same initial side and terminal side but have different measures are called angles. 66. The values of the trigonometric functions repeat infinetly at a specific interval, and so they are called, and this specific interval over which they repeat is called their. 67. The tangent function is when cos θ = 0. Therefor, its graph has as these locations. 68. The function and cosine function are reciprocals. 69. Since the sine function is odd, sin( θ) =.
70. What is the radius of the unit circle? 71. What is the equation of the unit circle? 72. One circle measures exactly 360 or exactly radians. 73. There are approximately radians in one circle. 74. There are approximately degrees in one radian. 75. There are exactly radians in 180. 76. sin A + B = 77. sin A B = 78. cos(a + B) = 79. cos A B = 80. tan A + B = 81. tan A B = 82. sin 2θ = 83. cos 2θ = or or 84. tan 2θ = 85. cos Q 4 = 86. sin Q 4 = 87. tan Q = or 4 88. If angle B lies in Quadrant III, then R lies in which quadrant? 4 89. Graph each of the following: a) y = arcsin x b) y = arccos x c) y = arctan x 90. If sin A = E and A lies in quadrant IV and sec B = 4 and B lies in quadrant I, find the exact @ value of each of the following:
a) sin(a + B) b) cos 2A c) tan(a B) d) cot T 4 e) cos(a + B) f) tan R 4 g) cos T 4 h) sin T 4 i) cos(b A) 91. Find the exact value of UVW 4XX YUVW 5X 5ZUVW 4XX UVW 5X 92. Find the exact value of cos 75 cos 15 + sin 75 sin 15 93. Find the exact value of each of the following: a) sin Z5 ( 1) b) sin Z5 1 c) cos Z5 5 4 d) sin Z5 5 4 e) cos Z5 4 4 i) tan Z5 E E f) cos Z5 (cos 6 54 ) g) cosz5 (cos( 46 E )) h) cos(cosz5 3.14)) j) arctan 3 k) sec Z5 ( 2) l) sec E6 4 m) sin Z5 (sin L6 [ )) n) cosz5 (sin 6 4 ) o) tanz5 (cos E6 7 ) p) tan(tanz5 4) q) cos Z5 (tan( \6 7 )) r) arcsin(sin π) s) tanz5 (sin E6 4 ) t) cos(sinz5 E 4 ) Recall: 94. Establish the identity: 5YUVW ] = tan x 5Y^_U ] 95. Establish the identity: `aw ] + 5Y^_` ] = 2 csc x 5Y^_` ] `aw ] 96. Establish the identity: UVW ]Y^_U ] `b^ ] ^`^ ] = 1 97. Establish the identity: 5Z`aW ] ^_` ] = ^_` ] 5Y`aW ] 98. Establish the identity: (1 cos 4 x)(1 + cot 4 x) = 1 99. Establish the identity: `b^ ] + `aw ] = 2 tan x ^`^ ] ^_` ] 100. Establish the identity: 1 `awc ] = cos x 5Y^_` ] 101. Suppose the terminal side of θ passes through the point (-2, 1), find θ to the nearest degree.
102. Suppose the terminal side of θ passes through the point E, 7, find θ to the nearest degree. @ @ 103. Suppose tan θ = 1.4. If θ lies in Quadrant I, what is its value to the nearest degree? If θ terminates in Quadrant 3, what is the value of θ to the nearest degree? 104. If cos A = 0.3, find sin A. 105. Suppose A and B are two acute angles in a right triangle. If cos A = 4 4, what is sin B? How do you know? 106. Convert 37 42 ' 17 to a decimal degree 107. Convert 5.23456 to DMS @ 108. The arc length of a circle is 6 cm and the radius is 5cm. What is the measure of the central angle in radians? (memorize s = rθ, where the central angle is measured in radians) 109. The arc length of a circle is 36 feet and the measure of the central angle is 90. What is the radius of the circle? 110. What is the y-intercept of y = sin(3x + π)? 111. What is the y-intercept of y = cos ] 4 + 6 7? 112. Given the graph below: a) Write a positive sine equation. b) Write a positive cosine equation. c) Write a negative sine equation. d) Write a negative cosine equation.
113. If I use my calculator to find the inverse sign of a fraction to determine the angle that corresponds to it, in what quadrant will the angle that my calculator gives me lie? In other words, if I do the inverse sine of a positive number, the calculator will output an angle in which quadrant? If I input the inverse sine of a negative number, the calculator will output an angle in which quadrant? Why is this? 114. If input is inverse cosine of a positive number, then the calculator outputs an angle in which quadrant? If input is inverse cosine of a negative number, then the calculator outputs an angle in which quadrant? 115. If input is inverse tangent of a positive number, then the calculator outputs an angle in which quadrant? If input is inverse tangent of a negative number, then the calculator outputs an angle in which quadrant? 116. Suppose sin A = \ 4@. a) Find the approximate measure of A to the nearest whole degree if A terminates in quadrant I. b) Find the approximate measure of A to the nearest whole degree if A terminates in quadrant II. c) Could A terminate in Q3 or Q4? Why or why not? d) If B is the other acute angle in a right triangle with A, what is cos B? How do you know? 117. If csc A = 5\ and A is in Quadrant III and sec B = 3 and terminates in Quadrant IV: [ a) Find sin(a B). Assume that A is in QII and B is in QIV. b) Find cos(a + B). Assume that A is in QII and B is in QIV. c) Find tan(a B). Assume that A is in QII and B is in QIV. d) Check your answers for a, b, c and show me how you know they are correct. Use decimals to do this. 118. If the ordered pair 5 4, E E lies on the terminal side of an angle in standard position, what is the exact measure of that angle in degrees and in radians? 119. What are the coordinates on the terminal side of θ if θ = E56 G? 120. What are the coordinates on the terminal side of θ if θ = 585?
121. If the terminal side of θ passes through the point ( 16, 4) what are the sine, cosine, tangent, cosecant, secant and cotangent of θ? 122. Find the exact value of each of the following without using a unit circle: a) sin 315 b) sin Z5 5 4 c) cos( 225 ) d) sin(sin Z5 4 4 ) e) tan Z5 ( 1) f) cos(sin Z5 4 4 ) g) tan(cos Z5 E 4 ) h) csc(tan Z5 1) i) cos Z5 (cos @6 7 ) j) sin Z5 (sin( \6 G )) 123. Simplify each of the following: a) cos 6 7 + θ b) sin θ + 6 G c) cos(270 θ) 124. If tan θ = 2.8 and θ lies in Q3, find the approximate measure of the angle to the nearest degree. 125. Suppose a right triangle has the legs measuring x and 3 1. The hypotenuse is 2 2. Find the simplest form of x that is completely rationalized. 126. If 0 < x < 90 and sin x = 5, what is the exact value of cos x? 4
127. You are going to ride a Ferris Wheel that is 3 feet above the ground and is rotating counter-clockwise at a rate of 1 revolution every 30 seconds. The diameter of the wheel is 50 feet. Write a negative cosine function that shows your height above the ground in terms of the number of seconds you have been on the ride. 128. Find the exact value of each of the following: a) sin 4 57 + cos 4 57 b) sin 80 cos 50 cos 80 sin 50 c) cos 20 cos 40 sin 20 sin 40 d) UVW 5GX ZUVW UVW 4@ 5YUVW 5GX UVW 4@ 129. If cos A = 4 4 (quadrant I) and sin B = 4 (quadrant 2), find: E E a) sec A i) cos T 4 b) sec ( A) j) cos 2B c) cos( A) k) sec(90 B) d) tan( B) l) sec(b 90 ) e) tan 2B m) csc(90 A) f) tan T 4 n) tan(a + B) g) cos R 4 h) sin 2A
130. A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the lake 90 feet from a piling that is directly across from the pier on the other side of the lake. From his transit, the angle between the piling and the pier is 35. What is the distance between the piling and the pier to the nearest foot? 131. What is the arc length from point A to point B in the diagram below? (Hint: remember s = rθ where θ is measured in radians and s is the arc length) 132. In a video game, the graphic of a butterfly needs to be rotated. To make the butterfly graphic rotate, the programmer uses the equations: x Š = x cos θ y sin θ y Š = x sin θ + y cos θ to transform each pixel of the graphic from its original coordinates (x, y) to its new coordinates (x Š, y Š ). Pixels may have negative or positive coordinates. If a pixel with coordinates (250, 100) is rotated by 6, what are the new coordinates? G