1. No category

Unit Circle – Class Work
Find the exact value of the given expression.
1. 𝑐𝑜𝑠
4𝜋
−
4. 𝑡𝑎𝑛
2. 𝑠𝑖𝑛
3
𝟏
𝟐
7𝜋
−
−5𝜋
5. 𝑐𝑜𝑡
6
3. 𝑠𝑒𝑐
4
4
6. 𝑐𝑠𝑐
−𝟏
√𝟑
𝟑
3 −2√10
7. Given the terminal point ( ,
7
𝐭𝐚𝐧 𝜽 = −
−9𝜋
2
−𝟏
) find tanθ
𝟐√𝟏𝟎
𝟑
8. Given the terminal point (
𝐜𝐨𝐭 𝜽 =
7
3
−𝟐
√𝟐
𝟐
15𝜋
2𝜋
−5 −12
13
,
13
) find cotθ
𝟓
𝟏𝟐
2
9. Knowing cosx= and the terminal point is in the fourth quadrant find sinx.
3
𝐬𝐢𝐧 𝒙 = −
√𝟓
𝟑
4
10. Knowing cotx= and the terminal point is in the third quadrant find secx.
5
𝐜𝐨𝐭 𝒙 = −
Pre-Calc Trig
𝟐√𝟏𝟎
𝟑
~1~
NJCTL.org
Unit Circle – Home Work
Find the exact value of the given expression.
11. 𝑐𝑜𝑠
5𝜋
12. 𝑠𝑖𝑛
3
𝟏
𝟐
14. 𝑡𝑎𝑛
4
13. 𝑠𝑒𝑐
15. 𝑐𝑜𝑡
6
13𝜋
4
16. 𝑐𝑠𝑐
𝟏
√𝟑
𝟑
7
−24
25
25
17. Given the terminal point ( ,
𝐜𝐨𝐭 𝜽 = −
3
−11𝜋
2
𝟏
) find cotθ
𝟕
𝟐𝟒
18. Given the terminal point (
𝐭𝐚𝐧 𝜽 = −
4𝜋
−𝟐
√𝟐
𝟐
−7𝜋
−
3𝜋
−4√2 7
9
, ) find tanθ
9
𝟕√𝟐
𝟖
7
19. Knowing sinx= and the terminal point is in the second quadrant find secx.
8
𝐬𝐞𝐜 𝒙 = −
20. Knowing cscx=
−4
5
𝟖√𝟏𝟓
𝟏𝟓
and the terminal point is in the third quadrant find cotx.
𝐜𝐨𝐭 𝒙 =
Pre-Calc Trig
𝟒
𝟑
~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
𝝅
A: 2 P: 𝝅 PS: − 𝟑 VS: 1
2
𝜋
3
6
A: 3 P:
23. 𝑦 = sin ( (𝑥 + )) + 3
𝝅
𝟐
PS:
𝝅
𝟒
VS: -2
24. 𝑦 = −1 cos(3𝑥 − 2𝜋) − 1
𝝅
A: 1 P: 𝟑𝝅 PS: − 𝟔 VS: 3
A: 1 P:
𝟐𝝅
𝟑
PS:
𝟐𝝅
𝟑
VS: -1
2
25. 𝑦 = cos(4𝑥 − 2𝜋) + 2
3
𝟐
A:𝟑 P:
Pre-Calc Trig
𝝅
𝟐
PS:
𝝅
𝟐
VS: 2
~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
A: 4 P: 𝟒𝝅 PS:
1
𝜋
4
2
27. 𝑦 = −2 cos(4𝑥 − 3𝜋) − 3
𝝅
VS: 2
𝟑
A: 2 P:
28. 𝑦 = 2 sin ( (𝑥 + )) + 1
𝝅
𝟐
PS:
𝟑𝝅
𝟒
VS: -3
29. 𝑦 = −1 cos(6𝑥 − 2𝜋) − 1
𝝅
A: 2 P: 𝟖𝝅 PS: − 𝟐 VS: 1
A: 1 P:
𝝅
𝟑
PS:
𝝅
𝟑
VS: 1
3
30. 𝑦 = cos(4𝑥 − 3𝜋) − 2
2
𝟑
𝝅
A: 𝟐 P: 𝟐 PS:
Pre-Calc Trig
𝟑𝝅
𝟒
VS: -2
~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
𝑪 = 𝟖𝟎𝒐 𝒂 = 𝟑. 𝟖𝟐 𝒃 = 𝟐. 𝟎𝟑
𝑨 = 𝟔𝟓𝒐 𝒃 = 𝟏𝟐 𝒄 = 𝟏𝟎. 𝟏𝟒
33. 𝑏 = 6, 𝐴 = 25°, 𝐵 = 45°
34. 𝑐 = 8, 𝐵 = 60°, 𝐶 = 40°
𝑪 = 𝟏𝟏𝟎𝒐 𝒂 = 𝟑. 𝟓𝟗 𝒄 = 𝟕. 𝟗𝟕
𝑨 = 𝟖𝟎𝒐 𝒂 = 𝟏𝟐. 𝟐𝟔 𝒃 = 𝟏𝟎. 𝟕𝟖
35. 𝑐 = 12, 𝑏 = 6, 𝐶 = 70°
36. 𝑏 = 12, 𝑎 = 15, 𝐵 = 40°
𝑨 = 𝟓𝟑. 𝟓𝒐 𝑪 = 𝟖𝟔. 𝟓𝒐 𝒄 = 𝟏𝟖. 𝟔𝟑
𝒐𝒓
𝒐
𝑨 = 𝟏𝟐𝟔. 𝟓 𝑪 = 𝟏𝟑. 𝟓𝒐 𝒄 = 𝟒. 𝟑𝟔
𝑨 = 𝟖𝟐𝒐 𝑩 = 𝟐𝟖𝒐 𝒂 = 𝟏𝟐. 𝟔𝟓
37. 𝐴 = 35°, 𝑎 = 6, 𝑏 = 11
𝑵𝒐 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏
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?
𝟏𝟑. 𝟕𝟑 𝒎𝒊𝒍𝒆𝒔 𝒇𝒓𝒐𝒎 𝑱𝑭𝑲
𝟗. 𝟒𝟏 𝒎𝒊𝒍𝒆𝒔 𝒊𝒏 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?
𝟑𝟓𝟏. 𝟓𝟕 𝒇𝒆𝒆𝒕 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
𝑨 = 𝟓𝟓𝒐 𝒃 = 𝟏𝟔. 𝟓𝟏 𝒃 = 𝟏𝟑. 𝟎𝟗
𝑪 = 𝟖𝟎𝒐 𝒂 = 𝟒. 𝟒 𝒃 = 𝟑. 𝟐𝟔
42. 𝑏 = 6, 𝐴 = 35°, 𝐵 = 45°
43. 𝑐 = 8, 𝐵 = 50°, 𝐶 = 40°
𝑪 = 𝟏𝟎𝟎𝒐 𝒂 = 𝟒. 𝟖𝟕 𝒄 = 𝟖. 𝟑𝟔
𝑨 = 𝟗𝟎𝒐 𝒂 = 𝟏𝟐. 𝟒𝟓 𝒃 = 𝟗. 𝟓𝟑
44. 𝑐 = 12, 𝑏 = 8, 𝐶 = 65°
45. 𝑏 = 12, 𝑎 = 16, 𝐵 = 50°
𝑵𝒐 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏
𝑨 = 𝟕𝟕. 𝟖𝒐 𝑩 = 𝟑𝟕. 𝟐𝒐 𝒂 = 𝟏𝟐. 𝟗𝟒
46. 𝐴 = 40°, 𝑎 = 5, 𝑏 = 12
𝑵𝒐 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏
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?
𝟏𝟓. 𝟖𝟖 𝒎𝒊𝒍𝒆𝒔 𝒇𝒓𝒐𝒎 𝑱𝑭𝑲
𝟏𝟐. 𝟑𝟒 𝒎𝒊𝒍𝒆𝒔 𝒊𝒏 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?
𝟖𝟏. 𝟗𝟔 𝒇𝒆𝒆𝒕 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
𝑨 = 𝟐𝟔. 𝟒𝒐 𝑩 = 𝟑𝟔. 𝟒𝒐 𝑪 = 𝟏𝟏𝟕. 𝟐𝒐
𝑨 = 𝟒𝟒. 𝟒𝒐 𝑩 = 𝟓𝟕. 𝟏𝒐 𝑪 = 𝟕𝟖. 𝟓𝒐
51. 𝑎 = 7, 𝑏 = 6, 𝑐 = 4
52. 𝐴 = 100°, 𝑏 = 4, 𝑐 = 5
𝒂 = 𝟔. 𝟗𝟐 𝑩 = 𝟑𝟒. 𝟕𝒐 𝑪 = 𝟒𝟓. 𝟑𝒐
𝑨 = 𝟖𝟔. 𝟒𝒐 𝑩 = 𝟓𝟖. 𝟖𝒐 𝑪 = 𝟑𝟒. 𝟖𝒐
53. 𝐵 = 60°, 𝑎 = 5, 𝑐 = 9
54. 𝐶 = 40°, 𝑎 = 10, 𝑏 = 12
𝑨 = 𝟑𝟑. 𝟕𝒐 𝒃 = 𝟕. 𝟖𝟏 𝑪 = 𝟖𝟔. 𝟑𝒐
𝑨 = 𝟓𝟔. 𝟑𝒐 𝑩 = 𝟖𝟑. 𝟕𝒐 𝒄 = 𝟕. 𝟕𝟔
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?
𝑻𝒉𝒆 𝒔𝒉𝒊𝒑 𝒊𝒔 𝟓𝟕𝟐𝟕. 𝟐𝟓 𝒇𝒕 𝒇𝒓𝒐𝒎 𝒍𝒊𝒈𝒉𝒕𝒉𝒐𝒖𝒔𝒆 𝑨
𝑻𝒉𝒆 𝒔𝒉𝒊𝒑 𝒊𝒔 𝟑𝟒𝟐𝟗. 𝟎𝟐 𝒇𝒕 𝒇𝒓𝒐𝒎 𝒍𝒊𝒈𝒉𝒕𝒉𝒐𝒖𝒔𝒆 𝑩
𝑻𝒉𝒆 𝒔𝒉𝒊𝒑 𝒊𝒔 𝟑𝟑𝟔𝟔. 𝟒 𝒇𝒕 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?
𝑻𝒉𝒆 𝒅𝒐𝒈𝒔 𝒂𝒓𝒆 𝟏𝟗. 𝟕 𝒎 𝒂𝒑𝒂𝒓𝒕
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
𝑨 = 𝟐𝟒. 𝟏𝒐 𝑩 = 𝟑𝟎. 𝟕𝒐 𝑪 = 𝟏𝟐𝟓. 𝟐𝒐
𝑨 = 𝟏𝟑. 𝟑𝒐 𝑩 = 𝟑𝟓. 𝟏𝒐 𝑪 = 𝟏𝟑𝟏. 𝟔𝒐
59. 𝑎 = 11, 𝑏 = 8, 𝑐 = 6
60. 𝐴 = 85°, 𝑏 = 3, 𝑐 = 7
𝑨 = 𝟏𝟎𝟐. 𝟔𝒐 𝑩 = 𝟒𝟓. 𝟐𝒐 𝑪 = 𝟑𝟐. 𝟐𝒐
𝒂 = 𝟕. 𝟑𝟕 𝑩 = 𝟐𝟑. 𝟗𝒐 𝑪 = 𝟕𝟏. 𝟏𝒐
61. 𝐵 = 70°, 𝑎 = 6, 𝑐 = 12
62. 𝐶 = 25°, 𝑎 = 14, 𝑏 = 19
𝑨 = 𝟐𝟗. 𝟔𝒐 𝒃 = 𝟏𝟏. 𝟒𝟑 𝑪 = 𝟖𝟎. 𝟒𝒐
𝑨 = 𝟖𝟔. 𝟖𝒐 𝑩 = 𝟔𝟖. 𝟐𝒐 𝒄 = 𝟖. 𝟔𝟓
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?
𝑻𝒉𝒆 𝒔𝒉𝒊𝒑 𝒊𝒔 𝟑𝟗𝟑𝟐. 𝟕 𝒇𝒕 𝒇𝒓𝒐𝒎 𝒍𝒊𝒈𝒉𝒕𝒉𝒐𝒖𝒔𝒆 𝑨
𝑻𝒉𝒆 𝒔𝒉𝒊𝒑 𝒊𝒔 𝟐𝟑𝟏𝟐. 𝟓 𝒇𝒕 𝒇𝒓𝒐𝒎 𝒍𝒊𝒈𝒉𝒕𝒉𝒐𝒖𝒔𝒆 𝑩
𝑻𝒉𝒆 𝒔𝒉𝒊𝒑 𝒊𝒔 𝟏𝟓𝟖𝟕. 𝟎 𝒇𝒕 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?
𝑻𝒉𝒆 𝒅𝒐𝒈𝒔 𝒂𝒓𝒆 𝟐𝟏. 𝟖 𝒎 𝒂𝒑𝒂𝒓𝒕
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
𝐬𝐞𝐜 𝒙
𝟏
68. (1 + cot 2 x)(1 − cos 2 x)
67. sin x (csc x − sin x)
𝟏
𝐜𝐨𝐬𝟐 𝒙
69. 1 −
tan2 x
70. (sin x − cos x)2
sec2 𝑥
𝐜𝐨𝐬𝟐 𝒙
71.
𝟏 − 𝐬𝐢𝐧 𝟐𝒙
cot2 x
72.
1−sin2 x
cosx
secx+tanx
𝟏 − 𝐬𝐢𝐧 𝒙
𝐜𝐬𝐜 𝟐 𝒙
73. sin 𝑥 tan 𝑥 + cos 𝑥
𝐬𝐞𝐜 𝒙
Verify the Identity
74. (1 − sin 𝑥)(1 + sin 𝑥) = cos 2 x
75.
tan 𝑥 cot 𝑥
sec 𝑥
= cos 𝑥
𝟏
𝐬𝐞𝐜 𝒙
𝟏 − 𝐬𝐢𝐧𝟐 𝒙
𝐜𝐨𝐬𝟐 𝒙
𝐜𝐨𝐬 𝒙
76. (1 − cos 2 x)(1 + tan2 x) = tan2 x
77.
(𝐬𝐢𝐧𝟐 𝒙)(𝐬𝐞𝐜 𝟐 𝒙)
(𝐬𝐢𝐧𝟐 𝒙) (
𝟏
𝐜𝐨𝐬 𝟐 𝒙
+
1
sec x−tan x
= 2 sec x
(𝐬𝐞𝐜 𝒙−𝐭𝐚𝐧 𝒙)+(𝐬𝐞𝐜 𝒙+𝐭𝐚𝐧 𝒙)
𝐬𝐞𝐜 𝟐 𝒙−𝐭𝐚𝐧𝟐 𝒙
)
𝟐 𝐬𝐞𝐜 𝒙
𝟏
𝐭𝐚𝐧𝟐 𝒙
Pre-Calc Trig
1
sec x+tan x
𝟐 𝐬𝐞𝐜 𝒙
~9~
NJCTL.org
Pythagorean Identities – Home Work
Simplify the expression
78. (tan x + cot x )2
79.
1−sin x
cos x
+
𝐬𝐞𝐜 𝟐 𝒙 + 𝐜𝐬𝐜 𝟐 𝒙
80.
cos x−cos y
sin x+sin y
+
𝟐 𝐬𝐞𝐜 𝒙
sin x−sin y
81.
cos x+cos y
1
sin 𝑥
−
𝟎
82.
cos x
1−sin x
1
csc 𝑥
𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙
1+sec2 x
83.
1+tan2 x
sin2 x
tan2 x
+
cos2 x
cot2 x
𝐜𝐨𝐬𝟐 𝒙 + 𝟏
84.
𝟏
𝑡𝑎𝑛2 𝑥
85.
1+𝑡𝑎𝑛2 𝑥
cos x
sec x
+
sin x
csc x
𝐬𝐢𝐧𝟐 𝒙
86.
1+sec2 x
1+tan2 x
+
𝟏
cos2 x
cot2 x
𝟐
Verify the Identity
87. 𝑐𝑜𝑠 2 𝑥 − 𝑠𝑖𝑛2 𝑥 = 1 − 2𝑠𝑖𝑛2 𝑥
88. tan 𝑥 cos 𝑥 csc 𝑥 = 1
(𝟏 − 𝐬𝐢𝐧𝟐 𝒙) − 𝐬𝐢𝐧𝟐 𝒙
𝐬𝐢𝐧 𝒙
𝟏
(𝐜𝐨𝐬 𝒙) (𝐜𝐨𝐬 𝒙) (𝐬𝐢𝐧 𝒙)
𝟏 − 𝟐 𝐬𝐢𝐧𝟐 𝒙
89.
1+cot x
csc x
𝟏
= sin x + cos x
90.
𝐜𝐨𝐬 𝒙
𝐬𝐢𝐧 𝒙
𝟏
𝐬𝐢𝐧 𝒙
cos x csc x
cot x
=1
𝟏
𝐬𝐢𝐧 𝒙
𝐜𝐨𝐬 𝒙
𝐬𝐢𝐧 𝒙
𝟏+
𝐜𝐨𝐬 𝒙
𝐜𝐨𝐬 𝒙
𝐜𝐨𝐬 𝒙
(𝟏 + 𝐬𝐢𝐧 𝒙 ) (
𝐬𝐢𝐧 𝒙
𝟏
𝐬𝐢𝐧 𝒙
)
𝐬𝐢𝐧 𝒙
∙ 𝐜𝐨𝐬 𝒙
𝟏
𝐬𝐢𝐧 𝒙 + 𝐜𝐨𝐬 𝒙
Pre-Calc Trig
~10~
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
√𝟔 + √𝟐
𝟒
√𝟔 − √𝟐
𝟒
93. tan 195
−𝟏 + √𝟑
12
√𝟐 − √𝟔
𝟒
𝟏 + √𝟑
95. cos
𝜋
94. 𝑠𝑖𝑛 −
19𝜋
96. 𝑡𝑎𝑛 −
12
𝜋
12
𝟏 − √𝟑
√𝟔 − √𝟐
𝟒
𝟏 + √𝟑
Verify the Identity.
𝜋
𝜋
3
3
𝝅
𝝅
𝝅
𝝅
𝟑
𝟑
𝟑
𝟑
𝟏
𝟏
𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 + 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 + 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 − 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧
𝜋
𝜋
1
4
4
2
98. cos (𝑥 + ) cos (𝑥 − ) = cos 2 𝑥 −
97. sin (𝑥 + ) + sin (𝑥 − ) = sin 𝑥
𝝅
𝝅
𝝅
𝝅
𝟒
𝟒
𝟒
𝟒
(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 − 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 ) (𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 + 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 )
𝐬𝐢𝐧 𝒙 ∗ 𝟐 + 𝐬𝐢𝐧 𝒙 ∗ 𝟐
(𝐜𝐨𝐬 𝒙 ∗
√𝟐
𝟐
− 𝐬𝐢𝐧 𝒙 ∗
𝟐
𝐬𝐢𝐧 𝒙
𝟒
𝟏
𝟐
√𝟐
𝟐
) (𝐜𝐨𝐬 𝒙 ∗
√𝟐
𝟐
+ 𝐬𝐢𝐧 𝒙 ∗
√𝟐
𝟐
𝟐
𝐜𝐨𝐬𝟐 𝒙 − 𝟒 𝐬𝐢𝐧𝟐 𝒙
𝟏
𝐜𝐨𝐬𝟐 𝒙 − 𝟐 (𝟏 − 𝐜𝐨𝐬𝟐 𝒙)
𝟏
𝐜𝐨𝐬𝟐 𝒙 − 𝟐
𝜋
tan 𝑥−1
4
tan 𝑥+1
99. tan (𝑥 − ) =
100.
𝝅
𝟒
𝝅
𝟏+𝐭𝐚𝐧 𝒙 𝐭𝐚𝐧
𝟒
sin(𝑥+𝑦)−sin(𝑥−𝑦)
cos(𝑥+𝑦)+cos(𝑥−𝑦)
= tan 𝑦
(𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚+𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚)−(𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚−𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚)
𝐭𝐚𝐧 𝒙−𝐭𝐚𝐧
(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚−𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒚)+(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚+𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒚)
𝟐 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚
𝐭𝐚𝐧 𝒙−𝟏
𝟐 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚
𝟏+𝐭𝐚𝐧 𝒙(𝟏)
𝐭𝐚𝐧 𝒚
Pre-Calc Trig
~11~
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
√𝟐 − √𝟔
𝟒
√𝟔 − √𝟐
𝟒
103.
tan 285
𝟏 − √𝟑
cos
12
√𝟐 − √𝟔
𝟒
𝟏 + √𝟑
105.
11𝜋
104. 𝑠𝑖𝑛 −
17𝜋
106. 𝑡𝑎𝑛 −
12
7𝜋
12
−𝟏 + √𝟑
√𝟐 − √𝟔
𝟒
𝟏 − √𝟑
Verify the Identity.
107.
sin (𝑥 +
(𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬
𝟐𝝅
𝟑
2𝜋
3
) + sin (𝑥 −
+ 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧
𝟐𝝅
𝟑
2𝜋
3
) = −sin 𝑥
) + (𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬
𝟏
𝟐𝝅
𝟑
− 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧
108. cos (𝑥 +
𝟐𝝅
𝟑
)
(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬
𝟑𝝅
𝟒
𝟏
𝐬𝐢𝐧 𝒙 ∗ − 𝟐 + 𝐬𝐢𝐧 𝒙 ∗ − 𝟐
(𝐜𝐨𝐬 𝒙 ∗
3𝜋
4
) cos (𝑥 −
− 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧
−√𝟐
𝟐
− 𝐬𝐢𝐧 𝒙
− 𝐬𝐢𝐧 𝒙 ∗
𝟐
𝟒
𝟑𝝅
𝟒
√𝟐
𝟐
3𝜋
4
) = cos 2 𝑥 −
) (𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬
) (𝐜𝐨𝐬 𝒙 ∗
𝟑𝝅
𝟒
−√𝟐
𝟐
1
2
+ 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧
+ 𝐬𝐢𝐧 𝒙 ∗
√𝟐
𝟐
𝟑𝝅
𝟒
)
)
𝟐
𝐜𝐨𝐬𝟐 𝒙 − 𝟒 𝐬𝐢𝐧𝟐 𝒙
𝟏
𝟏
𝐜𝐨𝐬𝟐 𝒙 − 𝟐 (𝟏 − 𝐜𝐨𝐬𝟐 𝒙)
𝟐
𝟏
𝐜𝐨𝐬𝟐 𝒙 − 𝟐
109.
tan (𝑥 +
5𝜋
4
)=
tan 𝑥+1
110. 𝑐𝑜𝑠 (
1−tan 𝑥
𝟓𝝅
𝟒
𝟓𝝅
𝟏−𝐭𝐚𝐧 𝒙 𝐭𝐚𝐧
𝟒
(𝐜𝐨𝐬
𝐭𝐚𝐧 𝒙+𝐭𝐚𝐧
𝟓𝝅
(
𝐭𝐚𝐧 𝒙+𝟏
𝟔
6
+ 𝑥) 𝑐𝑜𝑠 (
𝐜𝐨𝐬 𝒙 − 𝐬𝐢𝐧
−√𝟑
𝟐
5𝜋
𝟓𝝅
𝟔
𝟒
4
𝐬𝐢𝐧 𝒙) (𝐜𝐨𝐬
−√𝟑
𝟐
𝟐
𝟒
𝟑
3
− 𝑥) = − sin2 𝑥
𝟏
𝟑
𝟓𝝅
𝟔
𝐜𝐨𝐬 𝒙 + 𝐬𝐢𝐧
𝟓𝝅
𝟔
𝐬𝐢𝐧 𝒙)
𝟏
∗ 𝐜𝐨𝐬 𝒙 + ∗ 𝐬𝐢𝐧 𝒙)
𝟐
𝟏
𝐜𝐨𝐬𝟐 𝒙 − 𝐬𝐢𝐧𝟐 𝒙
𝟒
𝟏
(𝟏 − 𝐬𝐢𝐧𝟐 𝒙) − 𝟒 𝐬𝐢𝐧𝟐 𝒙
𝟑
𝟒
~12~
6
∗ 𝐜𝐨𝐬 𝒙 − ∗ 𝐬𝐢𝐧 𝒙) (
𝟏−𝐭𝐚𝐧 𝒙(𝟏)
Pre-Calc Trig
5𝜋
− 𝐬𝐢𝐧𝟐 𝒙
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
, 𝑓𝑖𝑛𝑑 tan 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
7
−3
7
−𝟕
𝟖
−√𝟏𝟓
𝟖
𝟏𝟐√𝟏𝟎
𝟑𝟏
, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
−𝟑𝟏
𝟒𝟗
, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
−𝟒𝟓
𝟓𝟑
−5
9
5
9
−
𝟒𝟓
𝟐𝟖
Verify the Identity.
117.
sin 3𝑥 = 3 sin 𝑥 − 4 sin3 𝑥
118. tan 3𝑥 =
3 tan 𝑥−𝑡𝑎𝑛3 𝑥
1−3𝑡𝑎𝑛2 𝑥
𝐬𝐢𝐧(𝟐𝒙 + 𝒙)
𝐭𝐚𝐧(𝟐𝒙 + 𝒙)
𝐬𝐢𝐧 𝟐𝒙 𝐜𝐨𝐬 𝒙 + 𝐜𝐨𝐬 𝟐𝒙 𝐬𝐢𝐧 𝒙
𝐭𝐚𝐧 𝟐𝒙+𝐭𝐚𝐧 𝒙
𝟏−𝐭𝐚𝐧 𝟐𝒙 𝐭𝐚𝐧 𝒙
(𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙) 𝐜𝐨𝐬 𝒙 + (𝟏 − 𝟐 𝐬𝐢𝐧𝟐 𝒙) 𝐬𝐢𝐧 𝒙
𝟐
𝟐 𝐭𝐚𝐧 𝒙
+𝐭𝐚𝐧 𝒙
𝟏−𝐭𝐚𝐧𝟐 𝒙
𝟐 𝐭𝐚𝐧 𝒙
𝟏−
𝐭𝐚𝐧 𝒙
𝟏−𝐭𝐚𝐧𝟐 𝒙
𝟑
𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙 + 𝐬𝐢𝐧 𝒙 − 𝟐 𝐬𝐢𝐧 𝒙
118.
𝟐 𝐬𝐢𝐧 𝒙 (𝟏 − 𝐬𝐢𝐧𝟐 𝒙) + 𝐬𝐢𝐧 𝒙 − 𝟐 𝐬𝐢𝐧𝟑 𝒙
𝟐 𝐭𝐚𝐧 𝒙+𝐭𝐚𝐧 𝒙−𝐭𝐚𝐧𝟑 𝒙
𝟏−𝐭𝐚𝐧𝟐 𝒙
𝟏−𝐭𝐚𝐧𝟐 𝒙−𝟐 𝐭𝐚𝐧𝟐 𝒙
𝟏−𝐭𝐚𝐧𝟐 𝒙
𝟑
𝟑 𝐬𝐢𝐧 𝒙 − 𝟒 𝐬𝐢𝐧 𝒙
119.
sin 4𝑥
sin 𝑥
= 4 cos 2𝑥 𝑐𝑜𝑠 𝑥
120. csc 2𝑥 =
𝟑 𝐭𝐚𝐧 𝒙−𝐭𝐚𝐧𝟑 𝒙
csc 𝑥
𝟏
𝐬𝐢𝐧 𝒙
𝐬𝐢𝐧 𝟐𝒙
𝟐(𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙) 𝐜𝐨𝐬 𝟐𝒙
𝟏
𝐬𝐢𝐧 𝒙
𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝟏
𝐬𝐢𝐧 𝒙
~13~
𝟏−𝟑 𝐭𝐚𝐧𝟐 𝒙
2 cos 𝑥
𝟐 𝐬𝐢𝐧 𝟐𝒙 𝐜𝐨𝐬 𝟐𝒙
𝟒 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝟐𝒙
Pre-Calc Trig
→
∗
𝟏
𝟐 𝐜𝐨𝐬 𝒙
→
𝐜𝐬𝐜 𝒙
𝟐 𝐜𝐨𝐬 𝒙
NJCTL.org
Double Angle Identity – Home Work
Find the exact value of the expression.
𝟏
𝟖
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
, 𝑓𝑖𝑛𝑑 tan 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
7
−5
7
−𝟑√𝟕
𝟖
−𝟐𝟎√𝟔
, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
−𝟏
𝟒𝟗
, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.
−𝟕𝟐
𝟗𝟕
−4
9
4
9
−
𝟕𝟐
𝟔𝟓
Verify the Identity.
127.
sec 2𝑥 =
sec2 𝑥
128.
2−sec2 𝑥
1+sin 2x
sin 2x
1
= 1 + sec x cscx
2
𝟏
𝟏
𝐜𝐨𝐬 𝟐𝒙
𝐬𝐢𝐧 𝟐𝒙
+𝟏
𝟏
𝟏
𝟐 𝐜𝐨𝐬𝟐 𝒙−𝟏
𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝟏
𝐜𝐨𝐬 𝟐 𝒙
𝟐 𝐜𝐨𝐬 𝟐 𝒙−𝟏
𝐜𝐨𝐬 𝟐 𝒙
𝟏
𝟐
+𝟏
𝐜𝐬𝐜 𝒙 𝐬𝐞𝐜 𝒙 + 𝟏
𝐬𝐞𝐜 𝟐 𝒙
𝟐−𝐬𝐞𝐜 𝟐 𝒙
129.
1 + cos 10𝑥 = 2 cos 2 5𝑥
𝟏 + (𝟐 𝐜𝐨𝐬 𝟐 𝟓𝒙 − 𝟏)
𝟐 𝐜𝐨𝐬 𝟐 𝟓𝒙
Pre-Calc Trig
~14~
NJCTL.org
Half Angle Identity – Class Work
Find the exact value of the expression.
130.
√
1−cos 6𝑥
𝐬𝐢𝐧 𝟑𝒙
132.
𝑥
𝑥
2
2
131. cos 2 ( ) − sin2 ( )
2
𝐜𝐨𝐬 𝟐𝒙
sin 22.5
133. tan 67.5
√𝟐
√𝟐 − √𝟐
𝟐
𝟐 − √𝟐
𝒐𝒓
𝟐 + √𝟐
𝟐
Verify the Identity.
134.
𝑥
2𝑡𝑎𝑛𝑥
2
tan 𝑥+sin 𝑥
sec = ±√
𝟐
𝐬𝐢𝐧 𝒙
𝟐 𝐬𝐢𝐧 𝒙
= √ 𝐬𝐢𝐧 𝒙𝐜𝐨𝐬 𝒙
+𝐬𝐢𝐧 𝒙
𝐜𝐨𝐬 𝒙
√
𝟐 𝐬𝐢𝐧 𝒙
𝐜𝐨𝐬 𝒙
𝐜𝐨𝐬 𝒙
→ √ 𝐬𝐢𝐧 𝒙+𝐬𝐢𝐧
∗ 𝐬𝐢𝐧 𝒙(𝟏+𝐜𝐨𝐬 𝒙) →
𝒙 𝐜𝐨𝐬 𝒙 → √
𝐜𝐨𝐬 𝒙
𝐜𝐨𝐬 𝒙
𝟐
𝟏+𝐜𝐨𝐬 𝒙
→
𝟏
𝟏+𝐜𝐨𝐬 𝒙
√
𝟐
→
𝟏
𝐜𝐨𝐬
𝒙
𝒙
𝟐
→ 𝐬𝐞𝐜 𝟐
Half Angle Identity – Home Work
Find the exact value of the expression.
135.
√
1+cos 4𝑥
𝐜𝐨𝐬 𝟐𝒙
137.
𝑥
𝑥
2
2
136. 2 cos ( ) sin ( )
2
𝐬𝐢𝐧 𝒙
cos 22.5
138. tan 15
𝟏
√𝟐 + √𝟐
𝟐
𝟐 + √𝟑
𝒐𝒓 𝟐 − √𝟑
Verify the Identity.
𝑥
139. tan = csc 𝑥 − cot 𝑥
2
=
Pre-Calc Trig
𝟏−𝐜𝐨𝐬 𝒙
𝐬𝐢𝐧 𝒙
→
𝟏
𝐬𝐢𝐧 𝒙
𝐜𝐨𝐬 𝒙
− 𝐬𝐢𝐧 𝒙 → 𝐜𝐬𝐜 𝒙 − 𝐜𝐨𝐭 𝒙
~15~
NJCTL.org
Power Reducing Identity – Class Work
Simplify the expression.
140. 𝑐𝑜𝑠 4 𝑥
𝟑
𝟖
𝟏
141. 𝑠𝑖𝑛8 𝑥
𝟏
𝟑𝟓
+ 𝟐 𝐜𝐨𝐬 𝟐𝒙 + 𝟖 𝐜𝐨𝐬 𝟒𝒙
𝟏𝟐𝟖
𝟏
𝟖
142.
𝟑
𝟏
𝐜𝐨𝐬 𝟐𝒙 𝐜𝐨𝐬 𝟒𝒙 + 𝟏𝟐𝟖 𝐜𝐨𝐬 𝟖𝒙
𝑠𝑖𝑛4 𝑥 𝑐𝑜𝑠 2 𝑥
𝟏
𝟏𝟔
143.
𝟑
− 𝟖 𝐜𝐨𝐬 𝟐𝒙 + 𝟑𝟐 𝐜𝐨𝐬 𝟒𝒙 −
𝟏
𝟏
𝟏
− 𝟏𝟔 𝐜𝐨𝐬 𝟐𝒙 − 𝟏𝟔 𝐜𝐨𝐬 𝟒𝒙 + 𝟏𝟔 𝐜𝐨𝐬 𝟐𝒙 𝐜𝐨𝐬 𝟒𝒙
𝜃
3
2
5
Find sin if cos 𝜃 = and 𝜃 is in the first quadrant.
√𝟓
𝟓
144.
𝜃
3
2
5
Find cos if tan 𝜃 = and 𝜃 is in the third quadrant.
−
Pre-Calc Trig
√𝟏𝟕(𝟑𝟒−𝟓√𝟑𝟒)
𝟑𝟒
~16~
NJCTL.org
Power Reducing Identity – Home Work
Simplify the expression.
145. 𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠 2 𝑥
𝟏
𝟖
146. 𝑠𝑖𝑛4 𝑥 𝑐𝑜𝑠 4 𝑥
𝟏
𝟑
− 𝟖 𝐜𝐨𝐬 𝟒𝒙
𝟏𝟐𝟖
𝟏
𝟏
− 𝟑𝟐 𝐜𝐨𝐬 𝟒𝒙 + 𝟏𝟐𝟖 𝐜𝐨𝐬 𝟖𝒙
𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠 4 𝑥
147.
𝟏
𝟏
𝟏
𝟏
+ 𝟏𝟔 𝐜𝐨𝐬 𝟐𝒙 − 𝟏𝟔 𝐜𝐨𝐬 𝟒𝒙 − 𝟏𝟔 𝐜𝐨𝐬 𝟐𝒙 𝐜𝐨𝐬 𝟒𝒙
𝟏𝟔
148.
𝜃
3
2
5
Find sin if cos 𝜃 = and 𝜃 is in the fourth quadrant.
−
149.
𝜃
−4
2
7
Find cos if sin 𝜃 =
√𝟓
and 𝜃 is in the third quadrant.
−
Pre-Calc Trig
𝟓
√𝟏𝟒(𝟕−√𝟑𝟑)
𝟏𝟒
~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
√𝟔
𝟐
−
152. cos 75 + cos 15
√𝟐
√𝟔
𝟐
𝟐
Verify the Identity.
153.
sin x+ sin5x
cos x+cos5x
= tan3x
154.
sin x + sin y
cos x−cos y
= − cot
x−y
155.
2
𝒙+𝒚
𝒙−𝒚
𝐜𝐨𝐬
𝟐
𝟐
𝒙+𝒚
𝒙−𝒚
−𝟐 𝐬𝐢𝐧
𝐬𝐢𝐧
𝟐
𝟐
𝟔𝒙
𝟒𝒙
𝟐 𝐬𝐢𝐧 𝟐 𝐜𝐨𝐬 𝟐
𝟔𝒙
𝟒𝒙
𝟐 𝐜𝐨𝐬 𝟐 𝐜𝐨𝐬 𝟐
𝐬𝐢𝐧 𝟑𝒙
𝐜𝐨𝐬 𝟑𝒙
𝒙−𝒚
𝟐
𝒙−𝒚
𝐬𝐢𝐧
𝟐
𝐜𝐨𝐬
−𝐜𝐨𝐭
𝟐
𝐜𝐨𝐭 𝒙
Sum to Product Identity – Home Work
Find the exact value of the expression.
156. sin 105 + sin 15
157. cos 105 – cos 15
−
= cot x
𝐜𝐨𝐬 𝒙
𝐬𝐢𝐧 𝒙
𝒙−𝒚
𝐭𝐚𝐧 𝟑𝒙
√𝟔
𝟐
sin 3x−sin x
𝟒𝒙
𝟐𝒙
𝟐 𝐜𝐨𝐬 𝟐 𝐜𝐨𝐬 𝟐
𝟐𝒙
𝟒𝒙
𝟐 𝐬𝐢𝐧 𝟐 𝐜𝐨𝐬 𝟐
𝟐 𝐬𝐢𝐧
−
cos x+cos 3x
158. cos 105 + cos 15
√𝟐
√𝟔
𝟐
𝟐
Verify the Identity.
159.
cos4x+cos2x
sin 4x+sin2x
= cot3x
160.
𝟐 𝐬𝐢𝐧
𝐜𝐨𝐬 𝟑𝒙
𝐬𝐢𝐧 𝟑𝒙
𝟐 𝐬𝐢𝐧 𝟑𝒙 𝐜𝐨𝐬(−𝟐𝒙) + 𝐬𝐢𝐧 𝟑𝒙
𝟐 𝐜𝐨𝐬 𝟑𝒙 𝐜𝐨𝐬(−𝟐𝒙) + 𝐜𝐨𝐬 𝟑𝒙
𝐜𝐨𝐭 𝟑𝒙
𝐬𝐢𝐧 𝟑𝒙 (𝟐 𝐜𝐨𝐬(−𝟐𝒙) + 𝟏)
𝐜𝐨𝐬 𝟑𝒙 (𝟐 𝐜𝐨𝐬(−𝟐𝒙) + 𝟏)
cos 87 + cos 33 = sin 63
𝟐 𝐜𝐨𝐬
𝟏𝟐𝟎
𝟐
𝐜𝐨𝐬
𝟓𝟒
𝟐
𝐬𝐢𝐧 𝟑𝒙
𝐜𝐨𝐬 𝟑𝒙
𝟐 𝐜𝐨𝐬 𝟔𝟎 𝐜𝐨𝐬 𝟐𝟕
𝐭𝐚𝐧 𝟑𝒙
𝟏
𝟐 ∗ 𝟐 ∗ 𝐬𝐢𝐧(𝟗𝟎 − 𝟐𝟕)
Pre-Calc Trig
𝐬𝐢𝐧 𝟔𝟑
= tan 3x
𝟔𝒙
−𝟒𝒙
𝐜𝐨𝐬
+ 𝐬𝐢𝐧 𝟑𝒙
𝟐
𝟐
𝟔𝒙
−𝟒𝒙
𝟐 𝐜𝐨𝐬 𝟐 𝐜𝐨𝐬 𝟐 + 𝐜𝐨𝐬 𝟑𝒙
𝟔𝒙
𝟐𝒙
𝐜𝐨𝐬
𝟐
𝟐
𝟔𝒙
𝟐𝒙
𝟐 𝐬𝐢𝐧 𝐜𝐨𝐬
𝟐
𝟐
𝟐 𝐜𝐨𝐬
161.
sin x+sin 5x+sin 3x
cos x+cos 5x+cos 3𝑥
~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
𝟏
√𝟐−√𝟑
𝟒
𝟒
2 sin 52.5 cos 97.5
165. 10 cos 6𝑥 sin 4𝑥
𝟓 𝐬𝐢𝐧 𝟏𝟎𝒙 − 𝟓 𝐬𝐢𝐧 𝟐𝒙
𝟏+√𝟐
𝟒
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
√𝟐+√𝟑
𝟏−√𝟑
𝟒
𝟒
4 cos 195 sin 15
169. 3 sin 8𝑥 cos 2𝑥
𝟑
𝟏
Pre-Calc Trig
𝟑
𝐬𝐢𝐧 𝟏𝟎𝒙 + 𝟐 𝐬𝐢𝐧 𝟔𝒙
𝟐
−𝟒
~19~
NJCTL.org
Inverse Trig Functions – Class Work
Evaluate the expression.
170.
171.
sin (𝑐𝑜𝑠 −1
5
13
6
170. 𝑐𝑜𝑠 (𝑡𝑎𝑛−1 − )
)
5
𝟏𝟐
𝟓√𝟔𝟏
𝟏𝟑
𝟔𝟏
3
𝑡𝑎𝑛 (𝑠𝑖𝑛−1 )
172. sin (𝑡𝑎𝑛−1 −
4
𝟑√𝟕
−
𝟕
173.
𝑐𝑜𝑠 (𝑠𝑖𝑛−1
6
11
𝟕√𝟐𝟏𝟖
𝟐𝟏𝟖
3
5
𝟒
−𝟑
𝟏𝟏
π
sin−1 (sin )
176. sin−1 (sin
4
𝝅
3π
4
)
𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
𝟒
177.
)
174. 𝑡𝑎𝑛 (𝑐𝑜𝑠 −1 − )
)
√𝟖𝟓
175.
7
13
π
π
cos −1 (cos )
178. cos −1 (cos − )
3
3
𝝅
𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
𝟑
Inverse Trig Functions – Home Work
Evaluate the expression.
179.
181.
sin (𝑐𝑜𝑠 −1
12
13
7
180. 𝑐𝑜𝑠 (𝑡𝑎𝑛−1 − )
)
5
𝟓
𝟓√𝟕𝟒
𝟏𝟑
𝟕𝟒
1
𝑡𝑎𝑛 (𝑠𝑖𝑛−1 )
182. sin (𝑡𝑎𝑛−1 −
4
√𝟏𝟓
−
𝟏𝟓
183.
𝑐𝑜𝑠 (𝑠𝑖𝑛−1
9
11
𝟓√𝟏𝟗𝟒
𝟏𝟗𝟒
4
5
𝟑
−𝟒
𝟏𝟏
π
sin−1 (sin )
186. sin−1 (sin
6
𝝅
cos −1 (cos
5π
6
)
𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
𝟔
187.
)
184. 𝑡𝑎𝑛 (𝑐𝑜𝑠 −1 − )
)
𝟐√𝟏𝟎
185.
5
13
2π
3
188. cos −1 (cos −
)
𝟐𝝅
2π
3
)
𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
𝟑
Pre-Calc Trig
~20~
NJCTL.org
Trig Equations – Class Work
Find the value(s) of x such that 0 ≤ 𝑥 < 2𝜋, if they exist.
189. sin 𝑥 = 1
𝒙=
191.
𝝅
190. 3 tan2 𝑥 = 1
𝝅 𝟓𝝅 𝟕𝝅 𝟏𝟏𝝅
𝒙 = 𝟔,
𝟐
𝑠𝑒𝑐 2 𝑥 − 2 = 0
193.
,
𝟒
,
𝟒
195.
𝟔
,
𝟔
,
𝒙=
𝟒
𝟐
𝟔
𝟔
,
𝟔
𝟑𝝅
𝟐
198. sin + cos 𝑥 = 0
2
𝝅 𝟑𝝅 𝟕𝝅 𝟏𝟏𝝅
,
,
𝑥
sin 2𝑥 + cos 𝑥 = 0
,
𝟔
196. 2(sin 𝑥 + 1) = 𝑐𝑜𝑠 2 𝑥
𝝅 𝟓𝝅
199.
𝟔
𝝅 𝟓𝝅 𝟕𝝅 𝟏𝟏𝝅
𝒙 = 𝟔,
𝟔
𝑠𝑖𝑛2 𝑥 − cos 𝑥 sin 𝑥 = 0
𝒙 = 𝟐,
𝟔
194. 3𝑠𝑒𝑐 2 𝑥 = 4
𝒙 = 𝟎, 𝝅, 𝟒 ,
197.
,
𝝅 𝟓𝝅
𝝅 𝟓𝝅 𝟕𝝅 𝟏𝟏𝝅
𝟔
𝟔
𝒙 = 𝟔,
𝟒
𝑐𝑠𝑐 2 𝑥 = 4
𝒙= ,
,
192. 2𝑠𝑖𝑛2 𝑥 + 3 = 7 sin 𝑥
𝝅 𝟑𝝅 𝟓𝝅 𝟕𝝅
𝒙 = 𝟒,
𝟔
𝝅 𝟓𝝅
𝒙 = 𝝅, 𝟑 ,
𝟔
𝟑
cos 2𝑥 + cos 𝑥 = 2
𝒙=𝟎
Pre-Calc Trig
~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
𝝅 𝟑𝝅 𝟓𝝅 𝟕𝝅
𝒙=𝝅
202.
201. 2 sin2 𝑥 = 1
𝒙 = 𝟒,
𝑐𝑠𝑐 2 𝑥 − 2 = 0
204.
𝟒
,
𝟒
,
𝒙=
𝟒
𝑠𝑒𝑐 2 𝑥 = 4
𝟑
206.
𝟑
,
𝟑
,
𝟑
208.
,𝟒,
𝒙=
𝟒
𝟑𝝅
𝟐
𝟑
,
𝟑
,
𝟑
𝟕𝝅 𝟏𝟏𝝅 𝝅
𝟔
,
𝟔
,𝟐
𝑥
sin 2𝑥 = 2tan 2𝑥
209. tan − sin 𝑥 = 0
2
𝝅 𝟑𝝅
𝒙 = 𝟎, 𝝅, 𝟐 ,
210.
𝟒
207. (sin 𝑥 − 1) = −2𝑐𝑜𝑠 2 𝑥
𝝅 𝟑𝝅 𝝅 𝟓𝝅
𝟐
,
𝝅 𝟐𝝅 𝟒𝝅 𝟓𝝅
𝒙= ,
𝟑
𝑐𝑜𝑠 2 𝑥 − cos 𝑥 sin 𝑥 = 0
𝒙 = 𝟐,
𝟒
205. 3𝑐𝑠𝑐 2 𝑥 = 4
𝝅 𝟐𝝅 𝟒𝝅 𝟓𝝅
𝒙= ,
,
203. 2𝑠𝑖𝑛2 𝑥 − 3 = sin 𝑥
𝝅 𝟑𝝅 𝟓𝝅 𝟕𝝅
𝒙 = 𝟒,
𝟒
𝝅 𝟑𝝅
𝒙 = 𝟎, 𝝅, 𝟐 ,
𝟐
𝟐
sin 2𝑥 − sin 𝑥 = 0
𝝅 𝟓𝝅
𝒙 = 𝟎, 𝝅, 𝟑 ,
Pre-Calc Trig
𝟑
~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. −
C
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
C
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
B
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
A
b. 2
c. 3
d. 8
Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 35°, 𝑎 = 5, & 𝑐 = 7, 𝑓𝑖𝑛𝑑 𝐵.
a. 18.418
b. 53.418
C
c. 91.582
d. both a and b
Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 50°, 𝑎 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.
a. 1.021
b. 40
D
c. 128.979
d. no solution
Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 50°, 𝑏 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.
a. 6.188
b. 32.456
C
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
D
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
A
2
10. On the interval [0, 2π), sin 2𝑥 = 0, thus x =
a. 0
π
b.
D
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.
B
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)
D
(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
D
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
C
, 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
B
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
I only
II and III
I and III
I, II, and III
III.
5π
3
D
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.
𝟏
𝒓 = 𝟑𝟐 𝒗𝟎 𝟐 𝐬𝐢𝐧 𝟐𝜽
b. A golf ball is hit 200 yards, if the initial velocity 200 ft/sec, what was the angle it was hit?
𝜽 = 𝟏𝟒. 𝟑𝟒𝒐
c.
If the golfer struck the ball at 45°, how far would the ball traveled?
𝒓 = 𝟏𝟐𝟓𝟎 𝒇𝒆𝒆𝒕
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?
𝟑𝟐𝟎. 𝟐 𝒇𝒆𝒆𝒕
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.)
𝟐𝟏𝟏. 𝟖 𝒇𝒆𝒆𝒕
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?
𝟑𝟕𝟖. 𝟖 𝒇𝒆𝒆𝒕
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?
𝟑𝟔𝟓
b. What is the average daily production for the year?
𝟒𝟓𝟔𝟔 𝒈𝒂𝒍𝒍𝒐𝒏𝒔
c.
Using the graph of M(d), what months during the year is production over 5500 gallons a day?
February thru May
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
Error is on line 4
Line 4 should read
𝟐 𝐜𝐨𝐬𝟐 𝒙 + 𝟐 𝐜𝐨𝐬 𝒙 = 𝟎
The rest of the problem is
𝟐 𝐜𝐨𝐬 𝒙 (𝐜𝐨𝐬 𝒙 + 𝟏) = 𝟎
𝟐 𝐜𝐨𝐬 𝒙 = 𝟎
𝐜𝐨𝐬 𝒙 + 𝟏 = 𝟎
𝐜𝐨𝐬 𝒙 = 𝟎
𝐜𝐨𝐬 𝒙 = −𝟏
𝝅
𝟑𝝅
𝒙 = , 𝝅,
𝟐
𝟐
Pre-Calc Trig
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NJCTL.org