Name:



Name: _________________________ Pre-Calculus

Semester One Exam Review

I. Second Nine Weeks

Graphing Trig Functions: sketch the graph of the function, identify the parts being asked.

1.[pic] 2. [pic]

Domain: ______ Range: ____________ Domain: ______ Range: _____________

Amplitude: ________ Period: __________ Amplitude: _______ Period: ___________

Phase shift: ____________ Vertical slide: __________ Phase shift:__________ Vertical slide: __________

3. [pic] 4. [pic]

a = _______ b = _______ Period: ________ a = _______ b = _______ Period: ________

Phase Shift: _______ V. Shift: _______ Phase shift: _______ V. Shift: _______

Asymptotes: ______________________ Asymptotes: _________________________

Domain: ______________Range: ______________ Domain: _______________ Range: __________

5. [pic] 6. [pic]

a = _______ b = _______ Period: ________ a = _______ b = _______ Period: ________

Phase Shift: _______ V. Shift: _______ Phase shift: _______ V. Shift: _______

Asymptotes: ______________________ Asymptotes: _________________________

Domain: ______________Range: ______________ Domain: _______________ Range: __________

7. [pic] 8. [pic]

Domain: ______ Range: ____________ Domain: ______ Range: _____________

Amplitude: ________ Period: __________ Amplitude: _______ Period: ___________

Phase shift: ____________ Vertical slide: __________ Phase shift:__________ Vertical slide: __________

Write the equation of the graph shown below.

9. 10.

11. 12.

Applications of trigonometric functions: Write the equation that expresses the following situations.

13. As you ride a Ferris wheel, your distance from the ground varies sinusoidally with time. You are in the last seat filled and the Ferris wheel starts immediately. Let t be the number of seconds that have elapsed since the Ferris wheel started. You find that it takes 5 seconds to reach the top, 57 feet above the ground, and that it makes a revolution every 10 seconds. The diameter of the wheel is 54 feet.

14. Han notices that Luke is not feeling well; he decides to take his temperature. His body temperature varies sinusoidally with time. 10 minutes after he started timing, it reached a high of 102[pic]F. 3 minutes later, it reaches its next low, 99[pic]F.

15. Virologists find that the number of people sick with a certain disease varies periodically. Assume that the number of sick people varies sinusoidally with time. Records stat being kept when t = 0 years. A minimum number, 50 people have the disease when t = 3.5 years. The next maximum, 250 people, occurred at t = 7.5 years.

Right Triangle Applications

16. Luke Skywalker’s “X-Wing” is coming in for a landing on Hoth. He is currently at an altitude of 23 km above the planets surface. The angle of depression from the plane to the ground is 6.2[pic]. What is Luke’s horizontal distance from the landing platform?

17. Bilbo Baggins has been offered a job to survey the height of a nearby building. From the base of the building, Bilbo measures a distance of 125 meters. From this distance, Bilbo is able to determine that the angle of elevation to the top of the cliff is 43[pic]. What is the height of the building?

18. An 1800-foot high cell phone tower is to be supported by guy wires running from the ground to the top. The angle of elevation from the ground to the top of the tower is [pic]. How long will each wire be? How far from the base of the tower must the wires be anchored in the ground?

19. In an emergency dive, Captain Nemo takes his submarine down. The angle of depression with the ocean’s surface is [pic]. If the sub goes for 1200 m along its downward path, how deep will it be? What horizontal distance from its starting point? In order to avoid detection from surface ships, the sub must dive to a depth of 2000m. How far must it travel along its downward path to reach a depth of 2000m?

Law of Sines/Cosines

20. Use the Law of Sines to find the missing parts of the triangles:

a. [pic] A = 75[pic], a = 18, b = 12 b. [pic]B = 100[pic], C = 28[pic], b = 13.2 c. [pic]B = 123[pic], b = 45, c = 60

21. Use the Law of Cosines to find the missing parts of the triangles:

a. a = 55, b = 25, c = 72 b. [pic] A = 56[pic], b = 27.3, c = 38 c. [pic]B = 10.5[pic], a = 40, c = 30

22. A surveyor measures three sides of a triangular field and gets 24 ft, 32 ft, and 18 ft. What is the measure of the largest angle of the triangle?

23. Hamlet is storming a castle and needs to find the height of the parapet in order to build scaffolding high enough to go over the walls. From where he is standing, he measures the angle of elevation to be 46[pic]. He moves 20 m closer to the castle and measures the angle of elevation to be 52[pic]. Find the height of the castle walls.

24. Two planes leave an airport at the same time. Plane A travels at 750 mph, while Plane B travels at 500 mph.

a. How far apart will they be after 5 hours if the angle between their paths is 75[pic]?

b. After 7 hours they are 1300 miles apart. What is the measure of the angle between their paths?

25. While on a secret mission to save Endor, Leia was put into space to check two satellites. As she approaches the satellites, she finds that one of them is 12 km from her and the other is 21 km. She measures the angle between the two satellites (Leia at the vertex) to be 95[pic]. How far apart are the satellites?

26. Bilbo and Frodo are standing 5 miles apart when they see a hot air balloon floating over their heads. Bilbo’s angle of elevation to the balloon is 14[pic] while Frodo’s angle of elevation to the balloon is 10[pic]. How high up is the balloon?

27. To avoid a deadly ion cloud, Captain Kirk must steer the Enterprise around it. He turns at an angle of 28[pic] from his original path. He flies for awhile then turns and intercepts his original path at an angle of 65[pic]. He ended up 7 light-years from when he left the path. How far did he travel in order to avoid the ion cloud?

Area of Triangles

Find the area of the following triangles:

a. a = 5, b = 9, [pic]C = 14[pic] b. b = 13, c = 21, [pic]A = 71[pic] c. a = 43, b = 27, c = 54 d. a = 21, b = 35, c = 27

28. Evaluate the following:

a. [pic] b. [pic] c. sin-1 [pic] d. cos(arccos –[pic])

e. cos (arcsin[pic]) f. arctan [pic] g. tan( arcsin[pic]) h. cos(arcsin 0)

Trigonometric Identities: Simplify the following.

29. [pic] 30. [pic] 31. [pic] 32. [pic]

33. [pic] 34. cos(-x) 35. [pic] 36. [pic]

37. tan( csc( 38. [pic][pic] 39. [pic] 40. [pic]

Verify Trigonometric Identities

41. 2cos2 x – sin2x + 1 = 3cos2x 42. sin x + cos xcot x = csc x 43. [pic]

44. [pic] 45. [pic] 46. [pic]

47. [pic] 48. [pic]

Simplify the following using the Trigonometric Identities.

49. [pic] 50. [pic] 51. [pic]

52. [pic] 53. [pic] 54. [pic]

55. [pic] 56. [pic] 57. [pic] 58. [pic]

59. [pic] 60. [pic] 61. [pic] 62. [pic]

63. [pic] 64. [pic] 65. [pic] 66. [pic] 67. [pic]

Simplify, then Evaluate.

68. sin75°cos15° + sin15°cos75° 69. [pic] 70. [pic]

71. [pic] 72. [pic] 73. [pic]

74. [pic] 75. [pic] 76. [pic] 77. [pic]

78. [pic] 79. [pic] 80. [pic]

Evaluate the exact value of each expression (No Calculators)

81. [pic] 82. [pic] 83. [pic] 84. [pic]

Solve the following:

85. [pic] 86. [pic] 87. [pic] 88. [pic]

89. [pic]

II. First Nine Weeks

Write the equation for the function that has the graph with the given characteristics.

90. Cubic translated 4 units right, reflected over the x-axis, and moved 2 units up.

91. Absolute value stretched horizontally by a scale factor of 3, and translated 4 units down.

92. Rational translated 5 units left, reflected over the x-axis, and translated 7 units up.

93. Quadratic reflected over the y-axis, stretched vertically by a factor of 3and shifted 1 unit down.

94. Square Root with a horizontal reflection, and shifted down 3 units.

95. Rational Function stretched horizontally by a factor of 2, translated 4 units left and 2 units up.

96. Absolute value translated right 1 units, reflected over the x-axis and stretched vertically by a factor of 5.

Determine if the following are Odd/Even/Neither. Also, determine if the function is symmetric about the origin/y-axis/neither.

97. [pic] 98. [pic] 99. [pic] 100. [pic] 101. [pic]

102. [pic] 103. [pic] 104. [pic] 105. [pic]

Graph the following functions.

106. [pic] 107. [pic] 108. [pic] 109. [pic]

Random questions.

110. Draw an even function that is not a parent function.

111. Draw a function such that[pic]. Is the function even/odd/neither? Symmetry?

112. Describe the symmetry of the function[pic].

Simplify the following for the functions below. List any and all restrictions.

Let: [pic] [pic] [pic] [pic]

113. [pic] 114. [pic] 115. [pic] 116. [pic] 117. [pic] 118. [pic]

119. [pic] 120. [pic] 121. [pic]

For 122- 126 Let: [pic] [pic] [pic]

122. [pic] 123. [pic] 124. [pic] 125. [pic] 126. [pic] 127. [pic]

Algebra of Functions: f(x) = 2x + 5 g(x) = [pic] h(x) = x2 – 4 j(x) = [pic]

128. Find f/g(x) and its domain 129. Find h(j(x)) and its domain

Inverse Functions:

Graph each function, restricting the Domain if necessary. Graph the inverse. Find the Inverse Algebraically.

130. [pic] 131. [pic] 132. [pic] 133. [pic]

134. [pic] 135. [pic] 136. [pic]

Piecewise Functions:

Graph the following:

137. [pic] 138.[pic]

139. [pic] 140. [pic]

Find the Domains of the following:

141. [pic] 142. [pic] 143. [pic] 144. [pic]

Graph the following, state which quadrant the angle lies in and find its Reference Angle.

145. [pic] 146. [pic] 147. [pic] 148. [pic] 149. [pic] 150. [pic]

151. A rotation of 740[pic] clockwise terminates in which quadrant.

152. Graph the angle that has a [pic] rotation counter-clockwise.

153. A sector of a circle with radius 5 has an arc length of 4. Find the measure of the central angle in radians.

154. A sector of a circle with radius 12 has an arc length of 17. Find the measure of the central angle in radians.

155. Determine the arc length of a circle of radius 9 cm intercepted by a central angle of [pic].

156. Determine the areas of a sector with the following information:

a. r = 5, [pic] = 120 [pic] b. r = 8.4, [pic] c. [pic]

Change the following to degrees or radians.

157. [pic] 158. [pic] 159. [pic] 160. [pic] 161. [pic] 162. [pic]

Find two coterminal angles for the following (one positive, one negative).

163. [pic] 164. [pic] 165. [pic] 166. [pic]

Find the exact values of the 6 trigonometric functions of the angle [pic].

167. [pic] 168. [pic] 169. [pic] 170. [pic]

Find [pic], [pic], and [pic] for the point (x, y) of the angle in standard positions.

171. (2, 5) 172. (4, 4) 173. (3, -10) 174. (-1, -4)

Evaluate the following, if possible. Give exact answers.

175. [pic] 176. [pic] 177. [pic] 178. [pic] 179. [pic] 180. [pic]

181. [pic] 182. [pic] 183. [pic] 184. [pic] 185. [pic] 186. [pic]

187. [pic] 188. [pic] 189. [pic] 190. [pic] 191. [pic] 192. [pic]

193. [pic] 194. [pic] 195. [pic]

Evaluate the following with a calculator. Round to three decimal places. Be sure to be in the correct mode.

196. [pic] 197. [pic] 198. [pic] 199. [pic] 200. [pic] 201. [pic]

202. [pic] 203. [pic] 204. [pic] 205. [pic]

206. Find the angle measure of the angle formed by [pic] rotation clockwise.

207. Is [pic] positive, negative, or zero?

208. If [pic] in Quadrant III, determine the exact value of [pic].

209. If [pic] in Quadrant IV, determine the exact value of [pic].

Determine the values of [pic] that makes each statement true from [pic].

210. [pic] 211. [pic] 212. [pic] 213. [pic] 214. [pic]

Determine the values of [pic] that makes each statement true from [pic].

215. [pic] 216. [pic] 217. [pic] 218. [pic] 219. [pic]

Find the values of [pic], where [pic]. Round to two decimal places. Remember there will be two answers for each question. (With Calculator)

220. [pic] 221. [pic] 222. [pic] 223. [pic] 224. [pic]

Find the values of [pic], where [pic]. Round to two decimal places. (With Calculator)

225. [pic] 226. [pic] 227. [pic] 228. [pic] 229. [pic]

-----------------------

−4

−2π

1

3π/2

3π/2

1

−2π

−4

−π

3

−2

−4

3π/2

1

−2π

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2π

x

y

−π

3

−2

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2π

x

y

3π/2

1

−2π

−4

−π

3

−2

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2π

x

y

3π/2

1

−2π

−4

−π

3

−2

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2

−π

3

−2

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2π

x

y

3π/2

1

−2π

−4

−π

3

−2

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2π

x

y

3π/2

1

−2π

−4

−π

3

−2

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2π

x

y

3π/2

1

−2π

−4

−π

3

−2

−4

π/2

−2π

4

2

−1

−3

4

2π

π

−π/2

−3π/2

2π

x

y

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