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Pre-Calc Trig ~1~ NJCTL.org
Unit Circle β Class Work
Find the exact value of the given expression.
1. πππ 4π
3 2. π ππ
7π
4 3. π ππ
2π
3
4. π‘ππβ5π
6 5. πππ‘
15π
4 6. ππ π
β9π
2
7. Given the terminal point (3
7,
β2β10
7) find tanΞΈ
8. Given the terminal point (β5
13,
β12
13) find cotΞΈ
9. Knowing cosx=2
3 and the terminal point is in the fourth quadrant find sinx.
10. Knowing cotx=4
5 and the terminal point is in the third quadrant find secx.
Pre-Calc Trig ~2~ NJCTL.org
Unit Circle β Home Work
Find the exact value of the given expression.
11. πππ 5π
3 12. π ππ
3π
4 13. π ππ
4π
3
14. π‘ππβ7π
6 15. πππ‘
13π
4 16. ππ π
β11π
2
17. Given the terminal point (7
25,
β24
25) find cotΞΈ
18. Given the terminal point (β4β2
9,
7
9) find tanΞΈ
19. Knowing sinx=7
8 and the terminal point is in the second quadrant find secx.
20. Knowing cscx=β4
5 and the terminal point is in the third quadrant find cotx.
Pre-Calc Trig ~3~ NJCTL.org
Graphing β Class Work
State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and
then check it with a graphing calculator.
21. π¦ = 2 cos (2 (π₯ +π
3)) + 1 22. π¦ = β3 cos(4π₯ β π) β 2
23. π¦ = sin (2
3(π₯ +
π
6)) + 3 24. π¦ = β1 cos(3π₯ β 2π) β 1
25. π¦ =2
3cos(4π₯ β 2π) + 2
Pre-Calc Trig ~4~ NJCTL.org
Graphing β Home Work
State the amplitude, period, phase shift, and vertical shift for each function. Draw the graph by hand and
then check it with a graphing calculator.
26. π¦ = β4 cos (1
2(π₯ β
π
3)) + 2 27. π¦ = β2 cos(4π₯ β 3π) β 3
28. π¦ = 2 sin (1
4(π₯ +
π
2)) + 1 29. π¦ = β1 cos(6π₯ β 2π) β 1
30. π¦ =3
2cos(4π₯ β 3π) β 2
Pre-Calc Trig ~5~ NJCTL.org
Law of Sines β Class Work
Solve triangle ABC.
31. π΄ = 70Β°, π΅ = 30Β°, π = 4 32. π΅ = 65Β°, πΆ = 50Β°, π = 12
33. π = 6, π΄ = 25Β°, π΅ = 45Β° 34. π = 8, π΅ = 60Β°, πΆ = 40Β°
35. π = 12, π = 6, πΆ = 70Β° 36. π = 12, π = 15, π΅ = 40Β°
37. π΄ = 35Β°, π = 6, π = 11
38. An airplane is on the radar at both Newark Liberty International and JFK airports that are 20 miles
apart. The angle of elevation from Newark to the plane is 42Β°and from JFK is 35Β° when the plane is
directly between them. How far is the plane from JFK? What is the planeβs elevation?
39. A mathematician walking in the woods noticed that the angle the angle of elevation to a bird at the
top of a tree is 50Β°, after walking 40β toward the tree, the angle is 55Β°. How far is she from the bird?
Pre-Calc Trig ~6~ NJCTL.org
Law of Sines β Home Work
Solve triangle ABC.
40. π΄ = 60Β°, π΅ = 40Β°, π = 5 41. π΅ = 75Β°, πΆ = 50Β°, π = 14
42. π = 6, π΄ = 35Β°, π΅ = 45Β° 43. π = 8, π΅ = 50Β°, πΆ = 40Β°
44. π = 12, π = 8, πΆ = 65Β° 45. π = 12, π = 16, π΅ = 50Β°
46. π΄ = 40Β°, π = 5, π = 12
47. An airplane is on the radar at both Newark Liberty International and JFK airports that are 20 miles
apart. The angle of elevation from Newark to the plane is 52Β°and from JFK is 45Β° when the plane is
directly between them. How far is the plane from JFK? What is the planeβs elevation?
48. A mathematician walking in the woods noticed that the angle the angle of elevation to a bird at the
top of a tree is 45Β°, after walking 30β toward the tree, the angle is 60Β°. How far is she from the bird?
Pre-Calc Trig ~7~ NJCTL.org
Law of Cosines β Class Work
Solve triangle ABC.
49. π = 3, π = 4, π = 6 50. π = 5, π = 6, π = 7
51. π = 7, π = 6, π = 4 52. π΄ = 100Β°, π = 4, π = 5
53. π΅ = 60Β°, π = 5, π = 9 54. πΆ = 40Β°, π = 10, π = 12
55. A ship at sea noticed two lighthouses that according to the charts are 1 mile apart. The light at
lighthouse A is 200β above sea level and the navigator on the ship measures the angle of elevation
to be 2Β°, how far is the ship from lighthouse A? The light at lighthouse B is 300β above sea level and
the navigator on the ship measures the angle of elevation to be 5Β°, how far is the ship from
lighthouse B? How far is the ship from shore?
56. A student takes his 2 dogs for a walk. He lets them off their leash in a field where Edison runs at 7
m/s and Einstein runs at 6 m/s. The student determines the angle between the dogs is 20Β°, how far
are the dogs from each other in 8 seconds?
Pre-Calc Trig ~8~ NJCTL.org
Law of Cosines β Home Work
Solve triangle ABC.
57. π = 4, π = 5, π = 8 58. π = 4, π = 10, π = 13
59. π = 11, π = 8, π = 6 60. π΄ = 85Β°, π = 3, π = 7
61. π΅ = 70Β°, π = 6, π = 12 62. πΆ = 25Β°, π = 14, π = 19
63. A ship at sea noticed two lighthouses that according to the charts are 1 mile apart. The light at
lighthouse A is 275β above sea level and the navigator on the ship measures the angle of elevation
to be 4Β°, how far is the ship from lighthouse A? The light at lighthouse B is 325β above sea level and
the navigator on the ship measures the angle of elevation to be 8Β°, how far is the ship from
lighthouse B? How far is the ship from shore?
64. A student takes his 2 dogs for a walk. He lets them off their leash in a field where Edison runs at 10
m/s and Einstein runs at 8 m/s. The student determines the angle between the dogs is 25Β°, how far
are the dogs from each other in 5 seconds?
Pre-Calc Trig ~9~ NJCTL.org
Pythagorean Identities β Class Work
Simplify the expression
65. csc π₯ tan π₯ 66. cot π₯ sec π₯ sin π₯
67. sin x (csc x β sin x) 68. (1 + cot2x)(1 β cos2x)
69. 1 βtan2x
sec2π₯ 70. (sin x β cos x)2
71. cot2x
1βsin2x 72.
cosx
secx+tanx
73. sin π₯ tan π₯ + cos π₯
Verify the Identity
74. (1 β sin π₯)(1 + sin π₯) = cos2 x 75. tan π₯ cot π₯
sec π₯= cos π₯
76. (1 β cos2x)(1 + tan2x) = tan2x 77. 1
sec x+tan x+
1
sec xβtan x= 2 sec x
Pre-Calc Trig ~10~ NJCTL.org
Pythagorean Identities β Home Work
Simplify the expression
78. (tan x + cot x )2 79. 1βsin x
cos x+
cos x
1βsin x
80. cos xβcos y
sin x+sin y+
sin xβsin y
cos x+cos y 81.
1
sin π₯β
1
csc π₯
82. 1+sec2x
1+tan2x 83.
sin2x
tan2x+
cos2x
cot2x
84. π‘ππ2π₯
1+π‘ππ2π₯ 85.
cos x
sec x+
sin x
csc x
86. 1+sec2x
1+tan2x+
cos2x
cot2x
Verify the Identity
87. πππ 2π₯ β π ππ2π₯ = 1 β 2π ππ2π₯ 88. tan π₯ cos π₯ csc π₯ = 1
89. 1+cot x
csc x= sin x + cos x 90.
cos x csc x
cot x= 1
Pre-Calc Trig ~11~ NJCTL.org
Angle Sum/Difference Identity β Class Work
Use Angle Sum/Difference Identity to find the exact value of the expression.
91. sin 105 92. cos 75
93. tan 195 94. π ππ βπ
12
95. cos19π
12 96. π‘ππ β
π
12
Verify the Identity.
97. sin (π₯ +π
3) + sin (π₯ β
π
3) = sin π₯ 98. cos (π₯ +
π
4) cos (π₯ β
π
4) = cos2 π₯ β
1
2
99. tan (π₯ βπ
4) =
tan π₯β1
tan π₯+1 100.
sin(π₯+π¦)βsin(π₯βπ¦)
cos(π₯+π¦)+cos(π₯βπ¦)= tan π¦
Pre-Calc Trig ~12~ NJCTL.org
Angle Sum/Difference Identity β Home Work
Use Angle Sum/Difference Identity to find the exact value of the expression.
101. sin 165 102. cos 105
103. tan 285 104. π ππ β11π
12
105. cos17π
12 106. π‘ππ β
7π
12
Verify the Identity.
107. sin (π₯ +2π
3) + sin (π₯ β
2π
3) = βsin π₯ 108. cos (π₯ +
3π
4) cos (π₯ β
3π
4) = cos2 π₯ β
1
2
109. tan (π₯ +5π
4) =
tan π₯+1
1βtan π₯ 110. πππ (
5π
6+ π₯) πππ (
5π
6β π₯) =
3
4β sin2 π₯
Pre-Calc Trig ~13~ NJCTL.org
Double Angle Identity β Class Work
Find the exact value of the expression.
111. πππ π =1
4, ππππ cos 2π ππ π ππ ππ π‘βπ ππππ π‘ ππ’ππππππ‘.
112. πππ π =1
4, ππππ sin 2π ππ π ππ ππ π‘βπ πππ’ππ‘β ππ’ππππππ‘.
113. π πππ =β3
7, ππππ tan 2π ππ π ππ ππ π‘βπ π‘βπππ ππ’ππππππ‘.
114. π πππ =β3
7, ππππ cos 2π ππ π ππ ππ π‘βπ πππ’ππ‘β ππ’ππππππ‘.
115. π‘πππ =β5
9, ππππ sin 2π ππ π ππ ππ π‘βπ π πππππ ππ’ππππππ‘.
116. πππ‘π =5
9, ππππ tan 2π ππ π ππ ππ π‘βπ π‘βπππ ππ’ππππππ‘.
Verify the Identity.
117. sin 3π₯ = 3 sin π₯ β 4 sin3 π₯ 118. tan 3π₯ =3 tan π₯βπ‘ππ3π₯
1β3π‘ππ2π₯
118.
119. sin 4π₯
sin π₯= 4 cos 2π₯ πππ π₯ 120. csc 2π₯ =
csc π₯
2 cos π₯
Pre-Calc Trig ~14~ NJCTL.org
Double Angle Identity β Home Work
Find the exact value of the expression.
121. πππ π =3
4, ππππ cos 2π ππ π ππ ππ π‘βπ ππππ π‘ ππ’ππππππ‘.
122. πππ π =3
4, ππππ sin 2π ππ π ππ ππ π‘βπ πππ’ππ‘β ππ’ππππππ‘.
123. π πππ =β5
7, ππππ tan 2π ππ π ππ ππ π‘βπ π‘βπππ ππ’ππππππ‘.
124. π πππ =β5
7, ππππ cos 2π ππ π ππ ππ π‘βπ πππ’ππ‘β ππ’ππππππ‘.
125. π‘πππ =β4
9, ππππ sin 2π ππ π ππ ππ π‘βπ π πππππ ππ’ππππππ‘.
126. πππ‘π =4
9, ππππ tan 2π ππ π ππ ππ π‘βπ π‘βπππ ππ’ππππππ‘.
Verify the Identity.
127. sec 2π₯ =sec2 π₯
2βsec2 π₯ 128.
1+sin 2x
sin 2x= 1 +
1
2sec x cscx
129. 1 + cos 10π₯ = 2 cos2 5π₯
Pre-Calc Trig ~15~ NJCTL.org
Half Angle Identity β Class Work
Find the exact value of the expression.
130. β1βcos 6π₯
2 131. cos2 (
π₯
2) β sin2 (
π₯
2)
132. sin 22.5 133. tan 67.5
Verify the Identity.
134. secπ₯
2= Β±β
2π‘πππ₯
tan π₯+sin π₯
Half Angle Identity β Home Work
Find the exact value of the expression.
135. β1+cos 4π₯
2 136. 2 cos (
π₯
2) sin (
π₯
2)
137. cos 22.5 138. tan 15
Verify the Identity.
139. tanπ₯
2= csc π₯ β cot π₯
Pre-Calc Trig ~16~ NJCTL.org
Power Reducing Identity β Class Work
Simplify the expression.
140. πππ 4π₯ 141. π ππ8π₯
142. π ππ4π₯ πππ 2π₯
143. Find sinπ
2 if cos π =
3
5 and π is in the first quadrant.
144. Find cosπ
2 if tan π =
3
5 and π is in the third quadrant.
Pre-Calc Trig ~17~ NJCTL.org
Power Reducing Identity β Home Work
Simplify the expression.
145. π ππ2π₯ πππ 2π₯ 146. π ππ4π₯ πππ 4π₯
147. π ππ2π₯ πππ 4π₯
148. Find sinπ
2 if cos π =
3
5 and π is in the fourth quadrant.
149. Find cosπ
2 if sin π =
β4
7 and π is in the third quadrant.
Pre-Calc Trig ~18~ NJCTL.org
Sum to Product Identity β Class Work
Find the exact value of the expression.
150. sin 75 + sin 15 151. cos 75 β cos 15 152. cos 75 + cos 15
Verify the Identity.
153. sin x+ sin5x
cos x+cos5x= tan3x 154.
sin x + sin y
cos xβcos y= β cot
xβy
2 155.
cos x+cos 3x
sin 3xβsin x= cot x
Sum to Product Identity β Home Work
Find the exact value of the expression.
156. sin 105 + sin 15 157. cos 105 β cos 15 158. cos 105 + cos 15
Verify the Identity.
159. cos4x+cos2x
sin 4x+sin2x= cot3x 160.
sin x+sin 5x+sin 3x
cos x+cos 5x+cos 3π₯= tan 3x
161. cos 87 + cos 33 = sin 63
Pre-Calc Trig ~19~ NJCTL.org
Product to Sum Identity β Class Work
Find the exact value of the expression.
162. cos 75 cos 15 163. sin 37.5 sin 7.5
164. 2 sin 52.5 cos 97.5 165. 10 cos 6π₯ sin 4π₯
Product to Sum Identity β Home Work
Find the exact value of the expression.
166. cos 37.5 cos 7.5 167. sin 45 sin 15
168. 4 cos 195 sin 15 169. 3 sin 8π₯ cos 2π₯
Pre-Calc Trig ~20~ NJCTL.org
Inverse Trig Functions β Class Work
Evaluate the expression.
170. sin (πππ β1 5
13) 170. πππ (π‘ππβ1 β
6
5)
171. π‘ππ (π ππβ1 3
4) 172. sin (π‘ππβ1 β
7
13)
173. πππ (π ππβ1 6
11) 174. π‘ππ (πππ β1 β
3
5)
175. sinβ1 (sinΟ
4) 176. sinβ1 (sin
3Ο
4)
177. cosβ1 (cosΟ
3) 178. cosβ1 (cos β
Ο
3)
Inverse Trig Functions β Home Work
Evaluate the expression.
179. sin (πππ β1 12
13) 180. πππ (π‘ππβ1 β
7
5)
181. π‘ππ (π ππβ1 1
4) 182. sin (π‘ππβ1 β
5
13)
183. πππ (π ππβ1 9
11) 184. π‘ππ (πππ β1 β
4
5)
185. sinβ1 (sinΟ
6) 186. sinβ1 (sin
5Ο
6)
187. cosβ1 (cos2Ο
3) 188. cosβ1 (cos β
2Ο
3)
Pre-Calc Trig ~21~ NJCTL.org
Trig Equations β Class Work
Find the value(s) of x such that 0 β€ π₯ < 2π, if they exist.
189. sin π₯ = 1 190. 3 tan2 π₯ = 1
191. π ππ2π₯ β 2 = 0 192. 2π ππ2π₯ + 3 = 7 sin π₯
193. ππ π2π₯ = 4 194. 3π ππ2π₯ = 4
195. π ππ2π₯ β cos π₯ sin π₯ = 0 196. 2(sin π₯ + 1) = πππ 2π₯
197. sin 2π₯ + cos π₯ = 0 198. sinπ₯
2+ cos π₯ = 0
199. cos 2π₯ + cos π₯ = 2
Pre-Calc Trig ~22~ NJCTL.org
Trig Equations β Home Work
Find the value(s) of x such that 0 β€ π₯ < 2π, if they exist.
200. cos π₯ = β1 201. 2 sin2 π₯ = 1
202. ππ π2π₯ β 2 = 0 203. 2π ππ2π₯ β 3 = sin π₯
204. π ππ2π₯ = 4 205. 3ππ π2π₯ = 4
206. πππ 2π₯ β cos π₯ sin π₯ = 0 207. (sin π₯ β 1) = β2πππ 2π₯
208. sin 2π₯ = 2tan 2π₯ 209. tanπ₯
2β sin π₯ = 0
210. sin 2π₯ β sin π₯ = 0
Pre-Calc Trig ~23~ NJCTL.org
Trigonometry Unit Review
Multiple Choice
1. Given the terminal point of (β2
2,
ββ2
2) find tan π.
a. Ο
4
b. βΟ
4
c. -1
d. 1
2. Knowing sec π₯ =β5
4 and the terminal point is in the second quadrant find cot π.
a. β4
5
b. 3
5
c. β4
3
d. β3
4
3. What is the phase shift of π¦ =5
3cos(6π₯ β 2π) + 3?
a. 1
2Ο
b. Ο
3
c. 1
3
d. 2π
4. The difference between the maximum of π¦ = 2 cos (2 (π₯ +π
3)) + 1 and π¦ = β3 cos(4π₯ β π) β 2 is
a. 1
b. 2
c. 3
d. 8
5. Given βπ΄π΅πΆ, π€ππ‘β π΄ = 35Β°, π = 5, & π = 7, ππππ π΅.
a. 18.418
b. 53.418
c. 91.582
d. both a and b
6. Given βπ΄π΅πΆ, π€ππ‘β π΄ = 50Β°, π = 6, & π = 8, ππππ π΅.
a. 1.021
b. 40
c. 128.979
d. no solution
7. Given βπ΄π΅πΆ, π€ππ‘β π΄ = 50Β°, π = 6, & π = 8, ππππ π΅.
a. 6.188
b. 32.456
c. 47.967
d. 82.033
8. (sec π₯ + tan π₯)(sec π₯ β tan π₯) =
a. 1 + 2 sec π₯ tan π₯
b. 1 β sec π₯ tan π₯
c. 1 β2 sin π₯
πππ 2π₯
d. 1
Pre-Calc Trig ~24~ NJCTL.org
9. Find the exact value of sinπ
12
a. β6ββ2
4
b. β6+β2
4
c. β6ββ2
2
d. β6ββ2
2
10. On the interval [0, 2Ο), sin 2π₯ = 0, thus x =
a. 0
b. Ο
2
c. 3Ο
2
d. all of the above
11. Find the exact value of cos 105
a. β2ββ3
2
b. ββ2ββ3
2
c. β2+β3
2
d. ββ2+β3
2
12. π ππ4π₯ =
a. 1
8(3 β cos π₯ + cos 4π₯)
b. 1
8(3 + cos π₯ + cos 4π₯)
c. 1
8(3 + 4 cos π₯ + cos 4π₯)
d. 1
8(3 β 4cos π₯ + cos 4π₯)
13. Rewrite cos 6π₯ sin 4π₯ as a sum or difference.
a. 1
2cos 10x β
1
2cos2x
b. 1
2cos 10x +
1
2cos2x
c. 1
2sin 10x β sin2x
d. 1
2sin 10x β
1
2sin2x
14. On the interval [0, 2Ο), sin 5π₯ + sin 3π₯ = 0
a. Ο
4
b. kΟ
4, where k β Integers
c. kΟ
4, where k β {0,1,2,6}
d. no solution on the interval given
15. π ππβ1 (sin4π
3) =
a. 4π
3
b. βπ
3
c. πππ‘β π πππ π
d. Undefined
Pre-Calc Trig ~25~ NJCTL.org
16. On the interval [0, 2Ο), solve 2sin2 π₯ + 3 cos π₯ = 3
I. 0 II. Ο
3 III.
5Ο
3
a. I only
b. II and III
c. I and III
d. I, II, and III
Extended Response
1. The range of a projectile launched at initial velocity π£0 and angle π, is
π =1
16π£0
2 sin π cos π,
where r is the horizontal distance, in feet, the projectile will travel.
a. Rewrite the formula using double angle formula.
b. A golf ball is hit 200 yards, if the initial velocity 200 ft/sec, what was the angle it was hit?
c. If the golfer struck the ball at 45Β°, how far would the ball traveled?
2. A state park hires a surveyor to map out the park.
a. A and B are on opposite sides of the lake, if the surveyor stands at point C and measures
angle ACB= 50 and CA= 400β and CB= 350β, how wide is the lake?
b. At a river the surveyor picks two spots, X and Y, on the same bank of the river and a tree, C,
on opposite bank. Angle X= 60 and angle Y= 50 and XY=300β, how wide is the river?
(Remember distance is measured along perpendiculars.)
c. The surveyor measured the angle to the top of a hill at the center of the park to be 32Β°. She
moved 200β closer and the angle to the top of the hill was 43Β°. How tall was the hill?
Pre-Calc Trig ~26~ NJCTL.org
3. The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by
π = 19.6 sin (2ππ
365+ 12.6) + 45
where d is the day, d=1 is January first.
a. What is the period of the function?
b. What is the average daily production for the year?
c. Using the graph of M(d), what months during the year is production over 5500 gallons a day?
4. A student was asked to solve the following equation over the interval [0, 2π). During his calculations
he might have made an error. Identify the error and correct his work so that he gets the right
answer.
cos π₯ + 1 = sin π₯
cos2x + 2 cos x + 1 = π ππ2π₯
cos2x + 2 cos x + 1 = 1 β πππ 2π₯
2 cos π₯ = 0
cos π₯ = 0
Ο
2,3Ο
2
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