Unit Circle – Class Work
Find the exact value of the given expression.
1. 𝑐𝑜𝑠
4. 𝑡𝑎𝑛
4𝜋
2. 𝑠𝑖𝑛
3
−5𝜋
5. 𝑐𝑜𝑡
6
3 −2√10
7. Given the terminal point ( ,
7
8. Given the terminal point (
7
−5 −12
13
,
13
7𝜋
3. 𝑠𝑒𝑐
4
15𝜋
4
6. 𝑐𝑠𝑐
2𝜋
3
−9𝜋
2
) find tanθ
) find cotθ
2
9. Knowing cosx= and the terminal point is in the fourth quadrant find sinx.
3
4
10. Knowing cotx= and the terminal point is in the third quadrant find secx.
5
Pre-Calc Trig
~1~
NJCTL.org
Unit Circle – Home Work
Find the exact value of the given expression.
11. 𝑐𝑜𝑠
14. 𝑡𝑎𝑛
5𝜋
12. 𝑠𝑖𝑛
3
−7𝜋
15. 𝑐𝑜𝑡
6
7
−24
25
25
17. Given the terminal point ( ,
18. Given the terminal point (
3𝜋
4
13𝜋
4
13. 𝑠𝑒𝑐
16. 𝑐𝑠𝑐
4𝜋
3
−11𝜋
2
) find cotθ
−4√2 7
9
, ) find tanθ
9
7
19. Knowing sinx= and the terminal point is in the second quadrant find secx.
8
20. Knowing cscx=
Pre-Calc Trig
−4
5
and the terminal point is in the third quadrant find cotx.
~2~
NJCTL.org
Graphing – 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.
𝜋
21. 𝑦 = 2 cos (2 (𝑥 + )) + 1
22. 𝑦 = −3 cos(4𝑥 − 𝜋) − 2
3
2
𝜋
3
6
23. 𝑦 = sin ( (𝑥 + )) + 3
24. 𝑦 = −1 cos(3𝑥 − 2𝜋) − 1
2
25. 𝑦 = cos(4𝑥 − 2𝜋) + 2
3
Pre-Calc Trig
~3~
NJCTL.org
Graphing – 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.
1
𝜋
2
3
26. 𝑦 = −4 cos ( (𝑥 − )) + 2
1
𝜋
4
2
27. 𝑦 = −2 cos(4𝑥 − 3𝜋) − 3
28. 𝑦 = 2 sin ( (𝑥 + )) + 1
29. 𝑦 = −1 cos(6𝑥 − 2𝜋) − 1
3
30. 𝑦 = cos(4𝑥 − 3𝜋) − 2
2
Pre-Calc Trig
~4~
NJCTL.org
Law of Sines – Class Work
Solve triangle ABC.
31. 𝐴 = 70°, 𝐵 = 30°, 𝑐 = 4
32. 𝐵 = 65°, 𝐶 = 50°, 𝑎 = 12
33. 𝑏 = 6, 𝐴 = 25°, 𝐵 = 45°
34. 𝑐 = 8, 𝐵 = 60°, 𝐶 = 40°
35. 𝑐 = 12, 𝑏 = 6, 𝐶 = 70°
36. 𝑏 = 12, 𝑎 = 15, 𝐵 = 40°
37. 𝐴 = 35°, 𝑎 = 6, 𝑏 = 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 42°and from JFK is 35° when the plane is
directly between them. How far is the plane from JFK? What is the plane’s 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 50°, after walking 40’ toward the tree, the angle is 55°. How far is she from the bird?
Pre-Calc Trig
~5~
NJCTL.org
Law of Sines – Home Work
Solve triangle ABC.
40. 𝐴 = 60°, 𝐵 = 40°, 𝑐 = 5
41. 𝐵 = 75°, 𝐶 = 50°, 𝑎 = 14
42. 𝑏 = 6, 𝐴 = 35°, 𝐵 = 45°
43. 𝑐 = 8, 𝐵 = 50°, 𝐶 = 40°
44. 𝑐 = 12, 𝑏 = 8, 𝐶 = 65°
45. 𝑏 = 12, 𝑎 = 16, 𝐵 = 50°
46. 𝐴 = 40°, 𝑎 = 5, 𝑏 = 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 52°and from JFK is 45° when the plane is
directly between them. How far is the plane from JFK? What is the plane’s 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 45°, after walking 30’ toward the tree, the angle is 60°. How far is she from the bird?
Pre-Calc Trig
~6~
NJCTL.org
Law of Cosines – Class Work
Solve triangle ABC.
49. 𝑎 = 3, 𝑏 = 4, 𝑐 = 6
50. 𝑎 = 5, 𝑏 = 6, 𝑐 = 7
51. 𝑎 = 7, 𝑏 = 6, 𝑐 = 4
52. 𝐴 = 100°, 𝑏 = 4, 𝑐 = 5
53. 𝐵 = 60°, 𝑎 = 5, 𝑐 = 9
54. 𝐶 = 40°, 𝑎 = 10, 𝑏 = 12
55. A ship at sea noticed two lighthouses that according to the charts are 1 mile apart. The light at
lighthouse A is 200’ above sea level and the navigator on the ship measures the angle of elevation
to be 2°, how far is the ship from lighthouse A? The light at lighthouse B is 300’ above sea level and
the navigator on the ship measures the angle of elevation to be 5°, 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 20°, how far
are the dogs from each other in 8 seconds?
Pre-Calc Trig
~7~
NJCTL.org
Law of Cosines – Home Work
Solve triangle ABC.
57. 𝑎 = 4, 𝑏 = 5, 𝑐 = 8
58. 𝑎 = 4, 𝑏 = 10, 𝑐 = 13
59. 𝑎 = 11, 𝑏 = 8, 𝑐 = 6
60. 𝐴 = 85°, 𝑏 = 3, 𝑐 = 7
61. 𝐵 = 70°, 𝑎 = 6, 𝑐 = 12
62. 𝐶 = 25°, 𝑎 = 14, 𝑏 = 19
63. A ship at sea noticed two lighthouses that according to the charts are 1 mile apart. The light at
lighthouse A is 275’ above sea level and the navigator on the ship measures the angle of elevation
to be 4°, how far is the ship from lighthouse A? The light at lighthouse B is 325’ above sea level and
the navigator on the ship measures the angle of elevation to be 8°, 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 25°, how far
are the dogs from each other in 5 seconds?
Pre-Calc Trig
~8~
NJCTL.org
Pythagorean Identities – Class Work
Simplify the expression
65. csc 𝑥 tan 𝑥
66. cot 𝑥 sec 𝑥 sin 𝑥
68. (1 + cot 2 x)(1 − cos 2 x)
67. sin x (csc x − sin x)
69. 1 −
71.
tan2 x
70. (sin x − cos x)2
sec2 𝑥
cot2 x
72.
1−sin2 x
cosx
secx+tanx
73. sin 𝑥 tan 𝑥 + cos 𝑥
Verify the Identity
74. (1 − sin 𝑥)(1 + sin 𝑥) = cos 2 x
75.
76. (1 − cos 2 x)(1 + tan2 x) = tan2 x
Pre-Calc Trig
77.
~9~
tan 𝑥 cot 𝑥
sec 𝑥
1
sec x+tan x
= cos 𝑥
+
1
sec x−tan x
= 2 sec x
NJCTL.org
Pythagorean Identities – Home Work
Simplify the expression
78. (tan x + cot x )2
80.
82.
84.
86.
cos x−cos y
sin x+sin y
+
79.
sin x−sin y
81.
cos x+cos y
1+sec2 x
83.
1+tan2 x
𝑡𝑎𝑛2 𝑥
85.
1+𝑡𝑎𝑛2 𝑥
1+sec2 x
1+tan2 x
+
1+cot x
csc x
1
sin 𝑥
−
sin2 x
tan2 x
cos x
sec x
+
1
csc 𝑥
+
+
cos x
1−sin x
cos2 x
cot2 x
sin x
csc x
cot2 x
88. tan 𝑥 cos 𝑥 csc 𝑥 = 1
= sin x + cos x
Pre-Calc Trig
cos x
cos2 x
Verify the Identity
87. 𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥 = 1 − 2𝑠𝑖𝑛2 𝑥
89.
1−sin x
90.
~10~
cos x csc x
cot x
=1
NJCTL.org
Angle Sum/Difference Identity – Class Work
Use Angle Sum/Difference Identity to find the exact value of the expression.
91. sin 105
92. cos 75
93. tan 195
95. cos
94. 𝑠𝑖𝑛 −
19𝜋
96. 𝑡𝑎𝑛 −
12
𝜋
12
𝜋
12
Verify the Identity.
𝜋
𝜋
3
3
𝜋
tan 𝑥−1
4
tan 𝑥+1
99. tan (𝑥 − ) =
Pre-Calc Trig
𝜋
𝜋
1
4
4
2
98. cos (𝑥 + ) cos (𝑥 − ) = cos 2 𝑥 −
97. sin (𝑥 + ) + sin (𝑥 − ) = sin 𝑥
100.
~11~
sin(𝑥+𝑦)−sin(𝑥−𝑦)
cos(𝑥+𝑦)+cos(𝑥−𝑦)
= tan 𝑦
NJCTL.org
Angle Sum/Difference Identity – Home Work
Use Angle Sum/Difference Identity to find the exact value of the expression.
101. sin 165
102. cos 105
103.
tan 285
105.
cos
11𝜋
104. 𝑠𝑖𝑛 −
17𝜋
12
106. 𝑡𝑎𝑛 −
12
7𝜋
12
Verify the Identity.
107.
sin (𝑥 +
109.
tan (𝑥 +
Pre-Calc Trig
2𝜋
3
) + sin (𝑥 −
5𝜋
4
)=
2𝜋
3
) = −sin 𝑥
108. cos (𝑥 +
tan 𝑥+1
110. 𝑐𝑜𝑠 (
1−tan 𝑥
~12~
5𝜋
6
3𝜋
4
) cos (𝑥 −
+ 𝑥) 𝑐𝑜𝑠 (
5𝜋
6
3𝜋
4
) = cos 2 𝑥 −
1
2
3
− 𝑥) = − sin2 𝑥
4
NJCTL.org
Double Angle Identity – Class Work
Find the exact value of the expression.
1
111.
𝑐𝑜𝑠𝜃 = , 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
112.
𝑐𝑜𝑠𝜃 = , 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
113.
𝑠𝑖𝑛𝜃 =
114.
𝑠𝑖𝑛𝜃 =
115.
𝑡𝑎𝑛𝜃 =
116.
𝑐𝑜𝑡𝜃 = , 𝑓𝑖𝑛𝑑 tan 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
4
1
4
−3
7
−3
7
, 𝑓𝑖𝑛𝑑 tan 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
−5
9
, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
5
9
Verify the Identity.
117.
sin 3𝑥 = 3 sin 𝑥 − 4 sin3 𝑥
118. tan 3𝑥 =
3 tan 𝑥−𝑡𝑎𝑛3 𝑥
1−3𝑡𝑎𝑛2 𝑥
118.
119.
sin 4𝑥
sin 𝑥
= 4 cos 2𝑥 𝑐𝑜𝑠 𝑥
Pre-Calc Trig
120. csc 2𝑥 =
~13~
csc 𝑥
2 cos 𝑥
NJCTL.org
Double Angle Identity – Home Work
Find the exact value of the expression.
3
121.
𝑐𝑜𝑠𝜃 = , 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
122.
𝑐𝑜𝑠𝜃 = , 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
123.
𝑠𝑖𝑛𝜃 =
124.
𝑠𝑖𝑛𝜃 =
125.
𝑡𝑎𝑛𝜃 =
126.
𝑐𝑜𝑡𝜃 = , 𝑓𝑖𝑛𝑑 tan 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
4
3
4
−5
7
−5
7
, 𝑓𝑖𝑛𝑑 tan 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
−4
9
, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
4
9
Verify the Identity.
sec2 𝑥
127.
sec 2𝑥 =
129.
1 + cos 10𝑥 = 2 cos 2 5𝑥
Pre-Calc Trig
128.
2−sec2 𝑥
~14~
1+sin 2x
sin 2x
1
= 1 + sec x cscx
2
NJCTL.org
Half Angle Identity – Class Work
Find the exact value of the expression.
1−cos 6𝑥
130.
√
132.
sin 22.5
𝑥
𝑥
2
2
131. cos 2 ( ) − sin2 ( )
2
133. tan 67.5
Verify the Identity.
134.
𝑥
2𝑡𝑎𝑛𝑥
2
tan 𝑥+sin 𝑥
sec = ±√
Half Angle Identity – Home Work
Find the exact value of the expression.
1+cos 4𝑥
135.
√
137.
cos 22.5
𝑥
𝑥
2
2
136. 2 cos ( ) sin ( )
2
138. tan 15
Verify the Identity.
𝑥
139. tan = csc 𝑥 − cot 𝑥
2
Pre-Calc Trig
~15~
NJCTL.org
Power Reducing Identity – Class Work
Simplify the expression.
140. 𝑐𝑜𝑠 4 𝑥
141. 𝑠𝑖𝑛8 𝑥
142.
𝑠𝑖𝑛4 𝑥 𝑐𝑜𝑠 2 𝑥
143.
Find sin if cos 𝜃 = and 𝜃 is in the first quadrant.
144.
Find cos if tan 𝜃 = and 𝜃 is in the third quadrant.
Pre-Calc Trig
𝜃
3
2
5
𝜃
3
2
5
~16~
NJCTL.org
Power Reducing Identity – Home Work
Simplify the expression.
145. 𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠 2 𝑥
146. 𝑠𝑖𝑛4 𝑥 𝑐𝑜𝑠 4 𝑥
147.
𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠 4 𝑥
148.
Find sin if cos 𝜃 = and 𝜃 is in the fourth quadrant.
149.
Find cos if sin 𝜃 =
Pre-Calc Trig
𝜃
3
2
5
𝜃
−4
2
7
and 𝜃 is in the third quadrant.
~17~
NJCTL.org
Sum to Product Identity – Class Work
Find the exact value of the expression.
150. sin 75 + sin 15
151. cos 75 – cos 15
152. cos 75 + cos 15
Verify the Identity.
153.
sin x+ sin5x
cos x+cos5x
= tan3x
154.
sin x + sin y
cos x−cos y
= − cot
x−y
2
Sum to Product Identity – Home Work
Find the exact value of the expression.
156. sin 105 + sin 15
157. cos 105 – cos 15
155.
cos x+cos 3x
sin 3x−sin x
= cot x
158. cos 105 + cos 15
Verify the Identity.
159.
161.
cos4x+cos2x
sin 4x+sin2x
= cot3x
160.
sin x+sin 5x+sin 3x
cos x+cos 5x+cos 3𝑥
= tan 3x
cos 87 + cos 33 = sin 63
Pre-Calc Trig
~18~
NJCTL.org
Product to Sum Identity – Class Work
Find the exact value of the expression.
162. cos 75 cos 15
164.
163. sin 37.5 sin 7.5
2 sin 52.5 cos 97.5
165. 10 cos 6𝑥 sin 4𝑥
Product to Sum Identity – Home Work
Find the exact value of the expression.
166. cos 37.5 cos 7.5
168.
167. sin 45 sin 15
4 cos 195 sin 15
Pre-Calc Trig
169. 3 sin 8𝑥 cos 2𝑥
~19~
NJCTL.org
Inverse Trig Functions – Class Work
Evaluate the expression.
5
170.
sin (𝑐𝑜𝑠 −1
171.
𝑡𝑎𝑛 (𝑠𝑖𝑛−1 )
173.
𝑐𝑜𝑠 (𝑠𝑖𝑛−1
175.
sin−1 (sin )
177.
cos −1 (cos )
13
6
170. 𝑐𝑜𝑠 (𝑡𝑎𝑛−1 − )
)
5
3
172. sin (𝑡𝑎𝑛−1 −
4
6
11
7
13
)
3
174. 𝑡𝑎𝑛 (𝑐𝑜𝑠 −1 − )
)
5
π
176. sin−1 (sin
4
π
3π
4
)
π
178. cos −1 (cos − )
3
3
Inverse Trig Functions – Home Work
Evaluate the expression.
12
179.
sin (𝑐𝑜𝑠 −1
181.
𝑡𝑎𝑛 (𝑠𝑖𝑛−1 )
183.
𝑐𝑜𝑠 (𝑠𝑖𝑛−1
185.
sin−1 (sin )
187.
cos −1 (cos
13
7
180. 𝑐𝑜𝑠 (𝑡𝑎𝑛−1 − )
)
5
1
182. sin (𝑡𝑎𝑛−1 −
4
9
11
4
5
π
186. sin−1 (sin
6
Pre-Calc Trig
3
)
184. 𝑡𝑎𝑛 (𝑐𝑜𝑠 −1 − )
)
2π
5
13
5π
6
)
188. cos −1 (cos −
)
~20~
2π
3
)
NJCTL.org
Trig Equations – Class Work
Find the value(s) of x such that 0 ≤ 𝑥 < 2𝜋, if they exist.
189. sin 𝑥 = 1
190. 3 tan2 𝑥 = 1
191.
𝑠𝑒𝑐 2 𝑥 − 2 = 0
192. 2𝑠𝑖𝑛2 𝑥 + 3 = 7 sin 𝑥
193.
𝑐𝑠𝑐 2 𝑥 = 4
194. 3𝑠𝑒𝑐 2 𝑥 = 4
195.
𝑠𝑖𝑛2 𝑥 − cos 𝑥 sin 𝑥 = 0
196. 2(sin 𝑥 + 1) = 𝑐𝑜𝑠 2 𝑥
197.
sin 2𝑥 + cos 𝑥 = 0
198. sin + cos 𝑥 = 0
199.
cos 2𝑥 + cos 𝑥 = 2
Pre-Calc Trig
𝑥
2
~21~
NJCTL.org
Trig Equations – Home Work
Find the value(s) of x such that 0 ≤ 𝑥 < 2𝜋, if they exist.
200. cos 𝑥 = −1
201. 2 sin2 𝑥 = 1
202.
𝑐𝑠𝑐 2 𝑥 − 2 = 0
203. 2𝑠𝑖𝑛2 𝑥 − 3 = sin 𝑥
204.
𝑠𝑒𝑐 2 𝑥 = 4
205. 3𝑐𝑠𝑐 2 𝑥 = 4
206.
𝑐𝑜𝑠 2 𝑥 − cos 𝑥 sin 𝑥 = 0
207. (sin 𝑥 − 1) = −2𝑐𝑜𝑠 2 𝑥
208.
sin 2𝑥 = 2tan 2𝑥
209. tan − sin 𝑥 = 0
210.
sin 2𝑥 − sin 𝑥 = 0
Pre-Calc Trig
𝑥
2
~22~
NJCTL.org
Trigonometry Unit Review
Multiple Choice
1. Given the terminal point of (
a.
√2 −√2
2
,
2
) find tan 𝜃.
π
4
b. −
π
4
c. -1
d. 1
2. Knowing sec 𝑥 =
a.
b.
c.
d.
−5
4
and the terminal point is in the second quadrant find cot 𝜃.
−4
5
3
5
−4
3
−3
4
5
3. What is the phase shift of 𝑦 = cos(6𝑥 − 2𝜋) + 3?
3
a.
b.
c.
1
2π
π
3
1
3
d. 2𝜋
𝜋
4. The difference between the maximum of 𝑦 = 2 cos (2 (𝑥 + )) + 1 and 𝑦 = −3 cos(4𝑥 − 𝜋) − 2 is
3
5.
6.
7.
8.
a. 1
b. 2
c. 3
d. 8
Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 35°, 𝑎 = 5, & 𝑐 = 7, 𝑓𝑖𝑛𝑑 𝐵.
a. 18.418
b. 53.418
c. 91.582
d. both a and b
Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 50°, 𝑎 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.
a. 1.021
b. 40
c. 128.979
d. no solution
Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 50°, 𝑏 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.
a. 6.188
b. 32.456
c. 47.967
d. 82.033
(sec 𝑥 + tan 𝑥)(sec 𝑥 − tan 𝑥) =
a. 1 + 2 sec 𝑥 tan 𝑥
b. 1 − sec 𝑥 tan 𝑥
c.
1−
2 sin 𝑥
𝑐𝑜𝑠 2 𝑥
d. 1
Pre-Calc Trig
~23~
NJCTL.org
9. Find the exact value of sin
a.
b.
c.
d.
𝜋
12
√6−√2
4
√6+√2
4
√6−√2
2
√6−√2
2
10. On the interval [0, 2π), sin 2𝑥 = 0, thus x =
a. 0
π
b.
c.
2
3π
2
d. all of the above
11. Find the exact value of cos 105
a.
√2−√3
2
√2−√3
b. −
c.
2
√2+√3
2
√2+√3
d. −
2
12. 𝑠𝑖𝑛4 𝑥 =
a.
b.
c.
d.
1
8
1
8
1
8
1
8
(3 − cos 𝑥 + cos 4𝑥)
(3 + cos 𝑥 + cos 4𝑥)
(3 + 4 cos 𝑥 + cos 4𝑥)
(3 − 4cos 𝑥 + cos 4𝑥)
13. Rewrite cos 6𝑥 sin 4𝑥 as a sum or difference.
a.
b.
c.
d.
1
2
1
2
1
2
1
2
1
cos 10x − cos2x
2
1
cos 10x + cos2x
2
sin 10x − sin2x
1
sin 10x − sin2x
2
14. On the interval [0, 2π), sin 5𝑥 + sin 3𝑥 = 0
π
a.
b.
c.
4
kπ
4
kπ
4
, where k ∈ Integers
, where k ∈ {0,1,2,6}
d. no solution on the interval given
15. 𝑠𝑖𝑛
−1
(sin
a.
4𝜋
3
)=
4𝜋
3
b. −
𝜋
3
c. 𝑏𝑜𝑡ℎ 𝑎 𝑎𝑛𝑑 𝑏
d. Undefined
Pre-Calc Trig
~24~
NJCTL.org
16. On the interval [0, 2π), solve 2sin2 𝑥 + 3 cos 𝑥 = 3
I. 0
a.
b.
c.
d.
II.
π
3
III.
5π
3
I only
II and III
I and III
I, II, and III
Extended Response
1. The range of a projectile launched at initial velocity 𝑣0 and angle 𝜃, is
𝑟=
1
𝑣 2
16 0
sin 𝜃 cos 𝜃,
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 45°, 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= 400’ and CB= 350’, 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=300’, 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 32°. She
moved 200’ closer and the angle to the top of the hill was 43°. How tall was the hill?
Pre-Calc Trig
~25~
NJCTL.org
3. The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by
2𝜋𝑑
𝑀 = 19.6 sin (
+ 12.6) + 45
365
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 𝑥 + 1 = sin 𝑥
cos 2 x + 2 cos x + 1 = 𝑠𝑖𝑛2 𝑥
cos 2 x + 2 cos x + 1 = 1 − 𝑐𝑜𝑠 2 𝑥
2 cos 𝑥 = 0
cos 𝑥 = 0
π 3π
,
2 2
Pre-Calc Trig
~26~
NJCTL.org