Unit Circle Class Work - Center For Teaching & Learningcontent.njctl.org/courses/math/pre-calculus/trigonometry-2/trigonometry-cw-hw... · Pre-Calc Trig ~1~ NJCTL.org Unit Circle

Pre-Calc Trig ~1~ NJCTL.org

Unit Circle – Class Work

Find the exact value of the given expression.

1. 𝑐𝑜𝑠4𝜋

3 2. 𝑠𝑖𝑛

7𝜋

4 3. 𝑠𝑒𝑐

2𝜋

3

4. 𝑡𝑎𝑛−5𝜋

6 5. 𝑐𝑜𝑡

15𝜋

4 6. 𝑐𝑠𝑐

−9𝜋

2

7. Given the terminal point (3

7,

−2√10

7) find tanθ

8. Given the terminal point (−5

13,

−12

13) find cotθ

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.

−𝟏

𝟐

−√𝟐

𝟐

−𝟐

√𝟑

𝟑

−𝟏 −𝟏

𝐭𝐚𝐧 𝜽 = −𝟐√𝟏𝟎

𝟑

𝐜𝐨𝐭 𝜽 =𝟓

𝟏𝟐

𝐬𝐢𝐧 𝒙 = −√𝟓

𝟑

𝐜𝐨𝐭 𝒙 = −𝟐√𝟏𝟎

𝟑


Unit Circle – Home Work

Find the exact value of the given expression.

11. 𝑐𝑜𝑠5𝜋

3 12. 𝑠𝑖𝑛

3𝜋

4 13. 𝑠𝑒𝑐

4𝜋

3

14. 𝑡𝑎𝑛−7𝜋

6 15. 𝑐𝑜𝑡

13𝜋

4 16. 𝑐𝑠𝑐

−11𝜋

2

17. Given the terminal point (7

25,

−24

25) find cotθ

18. Given the terminal point (−4√2

9,

7

9) find tanθ

19. Knowing sinx=7

8 and the terminal point is in the second quadrant find secx.

20. Knowing cscx=−4

5 and the terminal point is in the third quadrant find cotx.

𝟏

𝟐 √𝟐

𝟐

−𝟐

−√𝟑

𝟑

𝟏 𝟏

𝐜𝐨𝐭 𝜽 = −𝟕

𝟐𝟒

𝐭𝐚𝐧 𝜽 = −𝟕√𝟐

𝟖

𝐬𝐞𝐜 𝒙 = −𝟖√𝟏𝟓

𝟏𝟓

𝐜𝐨𝐭 𝒙 =𝟒

𝟑


A: 2 P: 𝝅 PS: −𝝅

𝟑 VS: 1 A: 3 P:

𝝅

𝟐 PS:

𝝅

𝟒 VS: -2

A: 1 P: 𝟑𝝅 PS: −𝝅

𝟔 VS: 3 A: 1 P:

𝟐𝝅

𝟑 PS:

𝟐𝝅

𝟑 VS: -1

A:𝟐

𝟑 P:

𝝅

𝟐 PS:

𝝅

𝟐 VS: 2

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.

21. 𝑦 = 2 cos (2 (𝑥 +𝜋

3)) + 1 22. 𝑦 = −3 cos(4𝑥 − 𝜋) − 2

23. 𝑦 = sin (2

3(𝑥 +

𝜋

6)) + 3 24. 𝑦 = −1 cos(3𝑥 − 2𝜋) − 1

25. 𝑦 =2

3cos(4𝑥 − 2𝜋) + 2


A: 4 P: 𝟒𝝅 PS: 𝝅

𝟑 VS: 2 A: 2 P:

𝝅

𝟐 PS:

𝟑𝝅

𝟒 VS: -3

A: 2 P: 𝟖𝝅 PS: −𝝅

𝟐 VS: 1 A: 1 P:

𝝅

𝟑 PS:

𝝅

𝟑 VS: 1

A: 𝟑

𝟐 P:

𝝅

𝟐 PS:

𝟑𝝅

𝟒 VS: -2

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.

26. 𝑦 = −4 cos (1

2(𝑥 −

𝜋

3)) + 2 27. 𝑦 = −2 cos(4𝑥 − 3𝜋) − 3

28. 𝑦 = 2 sin (1

4(𝑥 +

𝜋

2)) + 1 29. 𝑦 = −1 cos(6𝑥 − 2𝜋) − 1

30. 𝑦 =3

2cos(4𝑥 − 3𝜋) − 2


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?

𝑪 = 𝟖𝟎𝒐 𝒂 = 𝟑. 𝟖𝟐 𝒃 = 𝟐. 𝟎𝟑 𝑨 = 𝟔𝟓𝒐 𝒃 = 𝟏𝟐 𝒄 = 𝟏𝟎. 𝟏𝟒

𝑪 = 𝟏𝟏𝟎𝒐 𝒂 = 𝟑. 𝟓𝟗 𝒄 = 𝟕. 𝟗𝟕 𝑨 = 𝟖𝟎𝒐 𝒂 = 𝟏𝟐. 𝟐𝟔 𝒃 = 𝟏𝟎. 𝟕𝟖

𝑨 = 𝟖𝟐𝒐 𝑩 = 𝟐𝟖𝒐 𝒂 = 𝟏𝟐. 𝟔𝟓 𝑨 = 𝟓𝟑. 𝟓𝒐 𝑪 = 𝟖𝟔. 𝟓𝒐 𝒄 = 𝟏𝟖. 𝟔𝟑

𝒐𝒓

𝑨 = 𝟏𝟐𝟔. 𝟓𝒐 𝑪 = 𝟏𝟑. 𝟓𝒐 𝒄 = 𝟒. 𝟑𝟔

𝑵𝒐 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏

𝟏𝟑. 𝟕𝟑 𝒎𝒊𝒍𝒆𝒔 𝒇𝒓𝒐𝒎 𝑱𝑭𝑲

𝟗. 𝟒𝟏 𝒎𝒊𝒍𝒆𝒔 𝒊𝒏 𝒆𝒍𝒆𝒗𝒂𝒕𝒊𝒐𝒏

𝟑𝟓𝟏. 𝟓𝟕 𝒇𝒆𝒆𝒕 𝒇𝒓𝒐𝒎 𝒕𝒉𝒆 𝒃𝒊𝒓𝒅


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?

𝑪 = 𝟖𝟎𝒐 𝒂 = 𝟒. 𝟒 𝒃 = 𝟑. 𝟐𝟔 𝑨 = 𝟓𝟓𝒐 𝒃 = 𝟏𝟔. 𝟓𝟏 𝒃 = 𝟏𝟑. 𝟎𝟗

𝑪 = 𝟏𝟎𝟎𝒐 𝒂 = 𝟒. 𝟖𝟕 𝒄 = 𝟖. 𝟑𝟔 𝑨 = 𝟗𝟎𝒐 𝒂 = 𝟏𝟐. 𝟒𝟓 𝒃 = 𝟗. 𝟓𝟑

𝑨 = 𝟕𝟕. 𝟖𝒐 𝑩 = 𝟑𝟕. 𝟐𝒐 𝒂 = 𝟏𝟐. 𝟗𝟒 𝑵𝒐 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏

𝑵𝒐 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏

𝟏𝟓. 𝟖𝟖 𝒎𝒊𝒍𝒆𝒔 𝒇𝒓𝒐𝒎 𝑱𝑭𝑲

𝟏𝟐. 𝟑𝟒 𝒎𝒊𝒍𝒆𝒔 𝒊𝒏 𝒆𝒍𝒆𝒗𝒂𝒕𝒊𝒐𝒏

𝟖𝟏. 𝟗𝟔 𝒇𝒆𝒆𝒕 𝒇𝒓𝒐𝒎 𝒕𝒉𝒆 𝒃𝒊𝒓𝒅


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?

𝑨 = 𝟐𝟔. 𝟒𝒐 𝑩 = 𝟑𝟔. 𝟒𝒐 𝑪 = 𝟏𝟏𝟕. 𝟐𝒐

𝒂 = 𝟔. 𝟗𝟐 𝑩 = 𝟑𝟒. 𝟕𝒐 𝑪 = 𝟒𝟓. 𝟑𝒐

𝑨 = 𝟑𝟑. 𝟕𝒐 𝒃 = 𝟕. 𝟖𝟏 𝑪 = 𝟖𝟔. 𝟑𝒐 𝑨 = 𝟓𝟔. 𝟑𝒐 𝑩 = 𝟖𝟑. 𝟕𝒐 𝒄 = 𝟕. 𝟕𝟔

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

𝑻𝒉𝒆 𝒅𝒐𝒈𝒔 𝒂𝒓𝒆 𝟏𝟗. 𝟕 𝒎 𝒂𝒑𝒂𝒓𝒕

𝑨 = 𝟒𝟒. 𝟒𝒐 𝑩 = 𝟓𝟕. 𝟏𝒐 𝑪 = 𝟕𝟖. 𝟓𝒐

𝑨 = 𝟖𝟔. 𝟒𝒐 𝑩 = 𝟓𝟖. 𝟖𝒐 𝑪 = 𝟑𝟒. 𝟖𝒐


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?

𝑨 = 𝟐𝟒. 𝟏𝒐 𝑩 = 𝟑𝟎. 𝟕𝒐 𝑪 = 𝟏𝟐𝟓. 𝟐𝒐 𝑨 = 𝟏𝟑. 𝟑𝒐 𝑩 = 𝟑𝟓. 𝟏𝒐 𝑪 = 𝟏𝟑𝟏. 𝟔𝒐

𝑨 = 𝟏𝟎𝟐. 𝟔𝒐 𝑩 = 𝟒𝟓. 𝟐𝒐 𝑪 = 𝟑𝟐. 𝟐𝒐 𝒂 = 𝟕. 𝟑𝟕 𝑩 = 𝟐𝟑. 𝟗𝒐 𝑪 = 𝟕𝟏. 𝟏𝒐

𝑨 = 𝟐𝟗. 𝟔𝒐 𝒃 = 𝟏𝟏. 𝟒𝟑 𝑪 = 𝟖𝟎. 𝟒𝒐 𝑨 = 𝟖𝟔. 𝟖𝒐 𝑩 = 𝟔𝟖. 𝟐𝒐 𝒄 = 𝟖. 𝟔𝟓

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

𝑻𝒉𝒆 𝒅𝒐𝒈𝒔 𝒂𝒓𝒆 𝟐𝟏. 𝟖 𝒎 𝒂𝒑𝒂𝒓𝒕


Pythagorean Identities – Class Work

Simplify the expression

65. csc 𝑥 tan 𝑥 66. cot 𝑥 sec 𝑥 sin 𝑥

67. sin x (csc x − sin x) 68. (1 + cot2x)(1 − cos2x)

69. 1 −tan2x

sec2𝑥 70. (sin x − cos x)2

71. cot2x

1−sin2x 72.

cosx

secx+tanx

73. sin 𝑥 tan 𝑥 + cos 𝑥

Verify the Identity

74. (1 − sin 𝑥)(1 + sin 𝑥) = cos2 x 75. tan 𝑥 cot 𝑥

sec 𝑥= cos 𝑥

76. (1 − cos2x)(1 + tan2x) = tan2x 77. 1

sec x+tan x+

1

sec x−tan x= 2 sec x

𝐬𝐞𝐜 𝒙 𝟏

𝐜𝐨𝐬𝟐 𝒙 𝟏

𝐜𝐨𝐬𝟐 𝒙 𝟏 − 𝐬𝐢𝐧 𝟐𝒙

𝐜𝐬𝐜𝟐 𝒙 𝟏 − 𝐬𝐢𝐧 𝒙

𝐬𝐞𝐜 𝒙

𝟏 − 𝐬𝐢𝐧𝟐 𝒙

𝐜𝐨𝐬𝟐 𝒙

𝟏

𝐬𝐞𝐜 𝒙

𝐜𝐨𝐬 𝒙

(𝐬𝐢𝐧𝟐 𝒙)(𝐬𝐞𝐜𝟐 𝒙)

(𝐬𝐢𝐧𝟐 𝒙) (𝟏

𝐜𝐨𝐬𝟐 𝒙)

𝐭𝐚𝐧𝟐 𝒙

(𝐬𝐞𝐜 𝒙−𝐭𝐚𝐧 𝒙)+(𝐬𝐞𝐜 𝒙+𝐭𝐚𝐧 𝒙)

𝐬𝐞𝐜𝟐 𝒙−𝐭𝐚𝐧𝟐 𝒙

𝟐 𝐬𝐞𝐜 𝒙

𝟏

𝟐 𝐬𝐞𝐜 𝒙


Pythagorean Identities – Home Work

Simplify the expression

78. (tan x + cot x )2 79. 1−sin x

cos x+

cos x

1−sin x

80. cos x−cos y

sin x+sin y+

sin x−sin y

cos x+cos y 81.

1

sin 𝑥−

1

csc 𝑥

82. 1+sec2x

1+tan2x 83.

sin2x

tan2x+

cos2x

cot2x

84. 𝑡𝑎𝑛2𝑥

1+𝑡𝑎𝑛2𝑥 85.

cos x

sec x+

sin x

csc x

86. 1+sec2x

1+tan2x+

cos2x

cot2x

Verify the Identity

87. 𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥 = 1 − 2𝑠𝑖𝑛2𝑥 88. tan 𝑥 cos 𝑥 csc 𝑥 = 1

89. 1+cot x

csc x= sin x + cos x 90.

cos x csc x

cot x= 1

𝐬𝐞𝐜𝟐 𝒙 + 𝐜𝐬𝐜𝟐 𝒙 𝟐 𝐬𝐞𝐜 𝒙

𝟎 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙

𝐜𝐨𝐬𝟐 𝒙 + 𝟏 𝟏

𝐬𝐢𝐧𝟐 𝒙 𝟏

𝟐

(𝟏 − 𝐬𝐢𝐧𝟐 𝒙) − 𝐬𝐢𝐧𝟐 𝒙

𝟏 − 𝟐 𝐬𝐢𝐧𝟐 𝒙

(𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙) (𝐜𝐨𝐬 𝒙) (

𝟏

𝐬𝐢𝐧 𝒙)

𝟏

𝟏+𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙𝟏

𝐬𝐢𝐧 𝒙

(𝟏 +𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙) (

𝐬𝐢𝐧 𝒙

𝟏)

𝐬𝐢𝐧 𝒙 + 𝐜𝐨𝐬 𝒙

𝐜𝐨𝐬 𝒙𝟏

𝐬𝐢𝐧 𝒙𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙∙

𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙

𝟏


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 94. 𝑠𝑖𝑛 −𝜋

12

95. cos19𝜋

12 96. 𝑡𝑎𝑛 −

𝜋

12

Verify the Identity.

97. sin (𝑥 +𝜋

3) + sin (𝑥 −

𝜋

3) = sin 𝑥 98. cos (𝑥 +

𝜋

4) cos (𝑥 −

𝜋

4) = cos2 𝑥 −

1

2

99. tan (𝑥 −𝜋

4) =

tan 𝑥−1

tan 𝑥+1 100.

sin(𝑥+𝑦)−sin(𝑥−𝑦)

cos(𝑥+𝑦)+cos(𝑥−𝑦)= tan 𝑦

√𝟔 + √𝟐

𝟒

√𝟔 − √𝟐

𝟒

−𝟏 + √𝟑

𝟏 + √𝟑

√𝟐 − √𝟔

𝟒

√𝟔 − √𝟐

𝟒

𝟏 − √𝟑

𝟏 + √𝟑

𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬𝝅

𝟑+ 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧

𝝅

𝟑+ 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬

𝝅

𝟑− 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧

𝝅

𝟑

𝐬𝐢𝐧 𝒙 ∗𝟏

𝟐+ 𝐬𝐢𝐧 𝒙 ∗

𝟏

𝟐

𝐬𝐢𝐧 𝒙

(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬𝝅

𝟒− 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧

𝝅

𝟒) (𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬

𝝅

𝟒+ 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧

𝝅

𝟒)

(𝐜𝐨𝐬 𝒙 ∗√𝟐

𝟐− 𝐬𝐢𝐧 𝒙 ∗

√𝟐

𝟐) (𝐜𝐨𝐬 𝒙 ∗

√𝟐


√𝟐

𝟐)

𝟐

𝟒𝐜𝐨𝐬𝟐 𝒙 −

𝟐

𝟒𝐬𝐢𝐧𝟐 𝒙

𝟏

𝟐𝐜𝐨𝐬𝟐 𝒙 −

𝟏

𝟐(𝟏 − 𝐜𝐨𝐬𝟐 𝒙)

𝐜𝐨𝐬𝟐 𝒙 −𝟏

𝟐

𝐭𝐚𝐧 𝒙−𝐭𝐚𝐧𝝅

𝟒

𝟏+𝐭𝐚𝐧 𝒙 𝐭𝐚𝐧𝝅

𝟒

𝐭𝐚𝐧 𝒙−𝟏

𝟏+𝐭𝐚𝐧 𝒙(𝟏)

(𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚+𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚)−(𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚−𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚)

(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚−𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒚)+(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚+𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒚)

𝟐 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚

𝟐 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚

𝐭𝐚𝐧 𝒚


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 285 104. 𝑠𝑖𝑛 −11𝜋

12

105. cos17𝜋

12 106. 𝑡𝑎𝑛 −

7𝜋

12


107. sin (𝑥 +2𝜋

3) + sin (𝑥 −

2𝜋

3) = −sin 𝑥 108. cos (𝑥 +

3𝜋

4) cos (𝑥 −

3𝜋

4) = cos2 𝑥 −

1

2

109. tan (𝑥 +5𝜋

4) =

tan 𝑥+1

1−tan 𝑥 110. 𝑐𝑜𝑠 (

5𝜋

6+ 𝑥) 𝑐𝑜𝑠 (

5𝜋

6− 𝑥) =

3

4− sin2 𝑥

√𝟔 − √𝟐

𝟒

√𝟐 − √𝟔

𝟒

𝟏 + √𝟑

𝟏 − √𝟑

√𝟐 − √𝟔

𝟒

√𝟐 − √𝟔

𝟒

−𝟏 + √𝟑

𝟏 − √𝟑

(𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬𝟐𝝅

𝟑+ 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧

𝟐𝝅

𝟑) + (𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬

𝟐𝝅

𝟑− 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧

𝟐𝝅

𝟑)

𝐬𝐢𝐧 𝒙 ∗ −𝟏

𝟐+ 𝐬𝐢𝐧 𝒙 ∗ −

𝟏

𝟐

− 𝐬𝐢𝐧 𝒙

(𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬𝟑𝝅

𝟒− 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧

𝟑𝝅

𝟒) (𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬

𝟑𝝅

𝟒+ 𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧

𝟑𝝅

𝟒)

(𝐜𝐨𝐬 𝒙 ∗−√𝟐

𝟐− 𝐬𝐢𝐧 𝒙 ∗

√𝟐

𝟐) (𝐜𝐨𝐬 𝒙 ∗

−√𝟐


√𝟐

𝟐)

𝟐


𝟐


𝟏

𝟐𝐜𝐨𝐬𝟐 𝒙 −

𝟏

𝟐(𝟏 − 𝐜𝐨𝐬𝟐 𝒙)

𝐜𝐨𝐬𝟐 𝒙 −𝟏

𝟐

𝐭𝐚𝐧 𝒙+𝐭𝐚𝐧𝟓𝝅

𝟒

𝟏−𝐭𝐚𝐧 𝒙 𝐭𝐚𝐧𝟓𝝅

𝟒

𝐭𝐚𝐧 𝒙+𝟏

𝟏−𝐭𝐚𝐧 𝒙(𝟏)

(𝐜𝐨𝐬𝟓𝝅

𝟔𝐜𝐨𝐬 𝒙 − 𝐬𝐢𝐧

𝟓𝝅

𝟔𝐬𝐢𝐧 𝒙) (𝐜𝐨𝐬

𝟓𝝅

𝟔𝐜𝐨𝐬 𝒙 + 𝐬𝐢𝐧

𝟓𝝅

𝟔𝐬𝐢𝐧 𝒙)

(−√𝟑

𝟐∗ 𝐜𝐨𝐬 𝒙 −

𝟏

𝟐∗ 𝐬𝐢𝐧 𝒙) (

−√𝟑

𝟐∗ 𝐜𝐨𝐬 𝒙 +

𝟏

𝟐∗ 𝐬𝐢𝐧 𝒙)

𝟑


𝟏


𝟑

𝟒(𝟏 − 𝐬𝐢𝐧𝟐 𝒙) −

𝟏


𝟑

𝟒− 𝐬𝐢𝐧𝟐 𝒙


Double Angle Identity – Class Work

Find the exact value of the expression.

111. 𝑐𝑜𝑠𝜃 =1

4, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

112. 𝑐𝑜𝑠𝜃 =1

4, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

113. 𝑠𝑖𝑛𝜃 =−3

7, 𝑓𝑖𝑛𝑑 tan 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

114. 𝑠𝑖𝑛𝜃 =−3

7, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

115. 𝑡𝑎𝑛𝜃 =−5

9, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

116. 𝑐𝑜𝑡𝜃 =5



117. sin 3𝑥 = 3 sin 𝑥 − 4 sin3 𝑥 118. tan 3𝑥 =3 tan 𝑥−𝑡𝑎𝑛3𝑥

1−3𝑡𝑎𝑛2𝑥

118.

119. sin 4𝑥

sin 𝑥= 4 cos 2𝑥 𝑐𝑜𝑠 𝑥 120. csc 2𝑥 =

csc 𝑥

2 cos 𝑥

−𝟕

𝟖

−√𝟏𝟓

𝟖

𝟏𝟐√𝟏𝟎

𝟑𝟏

−𝟑𝟏

𝟒𝟗

−𝟒𝟓

𝟓𝟑

−𝟒𝟓

𝟐𝟖

𝐬𝐢𝐧(𝟐𝒙 + 𝒙)

𝐬𝐢𝐧 𝟐𝒙 𝐜𝐨𝐬 𝒙 + 𝐜𝐨𝐬 𝟐𝒙 𝐬𝐢𝐧 𝒙

(𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙) 𝐜𝐨𝐬 𝒙 + (𝟏 − 𝟐 𝐬𝐢𝐧𝟐 𝒙) 𝐬𝐢𝐧 𝒙

𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬𝟐 𝒙 + 𝐬𝐢𝐧 𝒙 − 𝟐 𝐬𝐢𝐧𝟑 𝒙

𝟐 𝐬𝐢𝐧 𝒙 (𝟏 − 𝐬𝐢𝐧𝟐 𝒙) + 𝐬𝐢𝐧 𝒙 − 𝟐 𝐬𝐢𝐧𝟑 𝒙

𝟑 𝐬𝐢𝐧 𝒙 − 𝟒 𝐬𝐢𝐧𝟑 𝒙

𝐭𝐚𝐧(𝟐𝒙 + 𝒙)

𝐭𝐚𝐧 𝟐𝒙+𝐭𝐚𝐧 𝒙

𝟏−𝐭𝐚𝐧 𝟐𝒙 𝐭𝐚𝐧 𝒙

𝟐 𝐭𝐚𝐧 𝒙

𝟏−𝐭𝐚𝐧𝟐 𝒙+𝐭𝐚𝐧 𝒙

𝟏−𝟐 𝐭𝐚𝐧 𝒙

𝟏−𝐭𝐚𝐧𝟐 𝒙𝐭𝐚𝐧 𝒙

𝟐 𝐭𝐚𝐧 𝒙+𝐭𝐚𝐧 𝒙−𝐭𝐚𝐧𝟑 𝒙

𝟏−𝐭𝐚𝐧𝟐 𝒙

𝟏−𝐭𝐚𝐧𝟐 𝒙−𝟐 𝐭𝐚𝐧𝟐 𝒙

𝟏−𝐭𝐚𝐧𝟐 𝒙

→𝟑 𝐭𝐚𝐧 𝒙−𝐭𝐚𝐧𝟑 𝒙

𝟏−𝟑 𝐭𝐚𝐧𝟐 𝒙

𝟐 𝐬𝐢𝐧 𝟐𝒙 𝐜𝐨𝐬 𝟐𝒙

𝐬𝐢𝐧 𝒙

𝟐(𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙) 𝐜𝐨𝐬 𝟐𝒙

𝐬𝐢𝐧 𝒙

𝟒 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝟐𝒙

𝟏

𝐬𝐢𝐧 𝟐𝒙

𝟏

𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙

𝟏

𝐬𝐢𝐧 𝒙∗

𝟏

𝟐 𝐜𝐨𝐬 𝒙→

𝐜𝐬𝐜 𝒙

𝟐 𝐜𝐨𝐬 𝒙


Double Angle Identity – Home Work


121. 𝑐𝑜𝑠𝜃 =3

4, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

122. 𝑐𝑜𝑠𝜃 =3

4, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

123. 𝑠𝑖𝑛𝜃 =−5


124. 𝑠𝑖𝑛𝜃 =−5

7, 𝑓𝑖𝑛𝑑 cos 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

125. 𝑡𝑎𝑛𝜃 =−4

9, 𝑓𝑖𝑛𝑑 sin 2𝜃 𝑖𝑓 𝜃 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡.

126. 𝑐𝑜𝑡𝜃 =4



127. sec 2𝑥 =sec2 𝑥

2−sec2 𝑥 128.

1+sin 2x

sin 2x= 1 +

1

2sec x cscx

129. 1 + cos 10𝑥 = 2 cos2 5𝑥

𝟏

𝟖

−𝟑√𝟕

𝟖

−𝟐𝟎√𝟔

−𝟏

𝟒𝟗

−𝟕𝟐

𝟗𝟕

−𝟕𝟐

𝟔𝟓

𝟏

𝐜𝐨𝐬 𝟐𝒙

𝟏

𝟐 𝐜𝐨𝐬𝟐 𝒙−𝟏

𝟏


𝟐 𝐜𝐨𝐬𝟐 𝒙−𝟏


𝐬𝐞𝐜𝟐 𝒙

𝟐−𝐬𝐞𝐜𝟐 𝒙

𝟏

𝐬𝐢𝐧 𝟐𝒙+ 𝟏

𝟏

𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙+ 𝟏

𝟏

𝟐𝐜𝐬𝐜 𝒙 𝐬𝐞𝐜 𝒙 + 𝟏

𝟏 + (𝟐 𝐜𝐨𝐬𝟐 𝟓𝒙 − 𝟏)

𝟐 𝐜𝐨𝐬𝟐 𝟓𝒙


Half Angle Identity – Class Work


130. √1−cos 6𝑥

2 131. cos2 (

𝑥

2) − sin2 (

𝑥

2)

132. sin 22.5 133. tan 67.5


134. sec𝑥

2= ±√

2𝑡𝑎𝑛𝑥

tan 𝑥+sin 𝑥

Half Angle Identity – Home Work


135. √1+cos 4𝑥

2 136. 2 cos (

𝑥

2) sin (

𝑥

2)

137. cos 22.5 138. tan 15


139. tan𝑥

2= csc 𝑥 − cot 𝑥

𝐬𝐢𝐧 𝟑𝒙 𝐜𝐨𝐬 𝟐𝒙

√𝟐 − √𝟐

𝟐

√𝟐

𝟐 − √𝟐 𝒐𝒓

𝟐 + √𝟐

𝟐

= √𝟐

𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙+𝐬𝐢𝐧 𝒙

→ √𝟐 𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙𝐬𝐢𝐧 𝒙+𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙

𝐜𝐨𝐬 𝒙

→ √𝟐 𝐬𝐢𝐧 𝒙

𝐜𝐨𝐬 𝒙∗

𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙(𝟏+𝐜𝐨𝐬 𝒙) →

√𝟐

𝟏+𝐜𝐨𝐬 𝒙 →

𝟏

√𝟏+𝐜𝐨𝐬 𝒙

𝟐

→ 𝟏

𝐜𝐨𝐬𝒙

𝟐

→ 𝐬𝐞𝐜𝒙

𝟐

𝐜𝐨𝐬 𝟐𝒙 𝐬𝐢𝐧 𝒙

√𝟐 + √𝟐

𝟐

𝟏

𝟐 + √𝟑 𝒐𝒓 𝟐 − √𝟑

=𝟏−𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙 →

𝟏

𝐬𝐢𝐧 𝒙−

𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙 → 𝐜𝐬𝐜 𝒙 − 𝐜𝐨𝐭 𝒙


Power Reducing Identity – Class Work

Simplify the expression.

140. 𝑐𝑜𝑠4𝑥 141. 𝑠𝑖𝑛8𝑥

142. 𝑠𝑖𝑛4𝑥 𝑐𝑜𝑠2𝑥

143. Find sin𝜃

2 if cos 𝜃 =

3

5 and 𝜃 is in the first quadrant.

144. Find cos𝜃

2 if tan 𝜃 =

3

5 and 𝜃 is in the third quadrant.

𝟑

𝟖+

𝟏

𝟐𝐜𝐨𝐬 𝟐𝒙 +

𝟏

𝟖𝐜𝐨𝐬 𝟒𝒙

𝟑𝟓

𝟏𝟐𝟖−

𝟑

𝟖𝐜𝐨𝐬 𝟐𝒙 +

𝟑

𝟑𝟐𝐜𝐨𝐬 𝟒𝒙 −

𝟏

𝟖𝐜𝐨𝐬 𝟐𝒙 𝐜𝐨𝐬 𝟒𝒙 +

𝟏

𝟏𝟐𝟖𝐜𝐨𝐬 𝟖𝒙

𝟏

𝟏𝟔−

𝟏

𝟏𝟔𝐜𝐨𝐬 𝟐𝒙 −

𝟏

𝟏𝟔𝐜𝐨𝐬 𝟒𝒙 +

𝟏

𝟏𝟔𝐜𝐨𝐬 𝟐𝒙 𝐜𝐨𝐬 𝟒𝒙

√𝟓

𝟓

−√𝟏𝟕(𝟑𝟒−𝟓√𝟑𝟒)

𝟑𝟒


Power Reducing Identity – Home Work

Simplify the expression.

145. 𝑠𝑖𝑛2𝑥 𝑐𝑜𝑠2𝑥 146. 𝑠𝑖𝑛4𝑥 𝑐𝑜𝑠4𝑥

147. 𝑠𝑖𝑛2𝑥 𝑐𝑜𝑠4𝑥

148. Find sin𝜃

2 if cos 𝜃 =

3

5 and 𝜃 is in the fourth quadrant.

149. Find cos𝜃

2 if sin 𝜃 =

−4

7 and 𝜃 is in the third quadrant.

𝟏

𝟖−

𝟏

𝟖𝐜𝐨𝐬 𝟒𝒙

𝟑

𝟏𝟐𝟖−

𝟏

𝟑𝟐𝐜𝐨𝐬 𝟒𝒙 +

𝟏

𝟏𝟐𝟖𝐜𝐨𝐬 𝟖𝒙

𝟏

𝟏𝟔+

𝟏

𝟏𝟔𝐜𝐨𝐬 𝟐𝒙 −

𝟏

𝟏𝟔𝐜𝐨𝐬 𝟒𝒙 −

𝟏

𝟏𝟔𝐜𝐨𝐬 𝟐𝒙 𝐜𝐨𝐬 𝟒𝒙

−√𝟓

𝟓

−√𝟏𝟒(𝟕−√𝟑𝟑)

𝟏𝟒


Sum to Product Identity – Class Work


150. sin 75 + sin 15 151. cos 75 – cos 15 152. cos 75 + cos 15


153. sin x+ sin5x

cos x+cos5x= tan3x 154.

sin x + sin y

cos x−cos y= − cot

x−y

2 155.

cos x+cos 3x

sin 3x−sin x= cot x

Sum to Product Identity – Home Work


156. sin 105 + sin 15 157. cos 105 – cos 15 158. cos 105 + cos 15


159. cos4x+cos2x

sin 4x+sin2x= cot3x 160.

sin x+sin 5x+sin 3x

cos x+cos 5x+cos 3𝑥= tan 3x

161. cos 87 + cos 33 = sin 63

√𝟔

𝟐

−√𝟐

𝟐

√𝟔

𝟐

√𝟔

𝟐

−√𝟔

𝟐

√𝟐

𝟐

𝟐 𝐬𝐢𝐧𝟔𝒙𝟐 𝐜𝐨𝐬

𝟒𝒙𝟐

𝟐 𝐜𝐨𝐬𝟔𝒙𝟐 𝐜𝐨𝐬

𝟒𝒙𝟐

𝐬𝐢𝐧 𝟑𝒙

𝐜𝐨𝐬 𝟑𝒙

𝐭𝐚𝐧 𝟑𝒙

𝟐 𝐬𝐢𝐧𝒙+𝒚

𝟐𝐜𝐨𝐬

𝒙−𝒚

𝟐

−𝟐 𝐬𝐢𝐧𝒙+𝒚

𝟐𝐬𝐢𝐧

𝒙−𝒚

𝟐

−𝐜𝐨𝐬

𝒙−𝒚

𝟐

𝐬𝐢𝐧𝒙−𝒚

𝟐

−𝐜𝐨𝐭𝒙−𝒚

𝟐

𝟐 𝐜𝐨𝐬𝟒𝒙𝟐 𝐜𝐨𝐬

𝟐𝒙𝟐

𝟐 𝐬𝐢𝐧𝟐𝒙𝟐 𝐜𝐨𝐬

𝟒𝒙𝟐

𝐜𝐨𝐬 𝒙

𝐬𝐢𝐧 𝒙

𝐜𝐨𝐭 𝒙

𝟐 𝐜𝐨𝐬𝟔𝒙

𝟐𝐜𝐨𝐬

𝟐𝒙

𝟐

𝟐 𝐬𝐢𝐧𝟔𝒙

𝟐𝐜𝐨𝐬

𝟐𝒙

𝟐



𝐜𝐨𝐭 𝟑𝒙

𝟐 𝐜𝐨𝐬𝟏𝟐𝟎

𝟐𝐜𝐨𝐬

𝟓𝟒

𝟐

𝟐 𝐜𝐨𝐬 𝟔𝟎 𝐜𝐨𝐬 𝟐𝟕

𝟐 ∗𝟏

𝟐∗ 𝐬𝐢𝐧(𝟗𝟎 − 𝟐𝟕)

𝐬𝐢𝐧 𝟔𝟑

𝟐 𝐬𝐢𝐧𝟔𝒙𝟐

𝐜𝐨𝐬−𝟒𝒙

𝟐+ 𝐬𝐢𝐧 𝟑𝒙

𝟐 𝐜𝐨𝐬𝟔𝒙𝟐 𝐜𝐨𝐬

−𝟒𝒙𝟐 + 𝐜𝐨𝐬 𝟑𝒙

𝟐 𝐬𝐢𝐧 𝟑𝒙 𝐜𝐨𝐬(−𝟐𝒙) + 𝐬𝐢𝐧 𝟑𝒙

𝟐 𝐜𝐨𝐬 𝟑𝒙 𝐜𝐨𝐬(−𝟐𝒙) + 𝐜𝐨𝐬 𝟑𝒙

𝐬𝐢𝐧 𝟑𝒙 (𝟐 𝐜𝐨𝐬(−𝟐𝒙) + 𝟏)

𝐜𝐨𝐬 𝟑𝒙 (𝟐 𝐜𝐨𝐬(−𝟐𝒙) + 𝟏)



𝐭𝐚𝐧 𝟑𝒙


Product to Sum Identity – Class Work


162. cos 75 cos 15 163. sin 37.5 sin 7.5

164. 2 sin 52.5 cos 97.5 165. 10 cos 6𝑥 sin 4𝑥

Product to Sum Identity – Home Work


166. cos 37.5 cos 7.5 167. sin 45 sin 15

168. 4 cos 195 sin 15 169. 3 sin 8𝑥 cos 2𝑥

√𝟐+√𝟑

𝟒

𝟏−√𝟑

𝟒

−𝟏

𝟒

𝟑

𝟐𝐬𝐢𝐧 𝟏𝟎𝒙 +

𝟑

𝟐𝐬𝐢𝐧 𝟔𝒙

𝟏

𝟒

√𝟐−√𝟑

𝟒

𝟏+√𝟐

𝟒

𝟓 𝐬𝐢𝐧 𝟏𝟎𝒙 − 𝟓 𝐬𝐢𝐧 𝟐𝒙


Inverse Trig Functions – Class Work

Evaluate the expression.

170. sin (𝑐𝑜𝑠−1 5

13) 170. 𝑐𝑜𝑠 (𝑡𝑎𝑛−1 −

6

5)

171. 𝑡𝑎𝑛 (𝑠𝑖𝑛−1 3

4) 172. sin (𝑡𝑎𝑛−1 −

7

13)

173. 𝑐𝑜𝑠 (𝑠𝑖𝑛−1 6

11) 174. 𝑡𝑎𝑛 (𝑐𝑜𝑠−1 −

3

5)

175. sin−1 (sinπ

4) 176. sin−1 (sin

3π

4)

177. cos−1 (cosπ

3) 178. cos−1 (cos −

π

3)

Inverse Trig Functions – Home Work

Evaluate the expression.

179. sin (𝑐𝑜𝑠−1 12

13) 180. 𝑐𝑜𝑠 (𝑡𝑎𝑛−1 −

7

5)

181. 𝑡𝑎𝑛 (𝑠𝑖𝑛−1 1

4) 182. sin (𝑡𝑎𝑛−1 −

5

13)

183. 𝑐𝑜𝑠 (𝑠𝑖𝑛−1 9

11) 184. 𝑡𝑎𝑛 (𝑐𝑜𝑠−1 −

4

5)

185. sin−1 (sinπ

6) 186. sin−1 (sin

5π

6)

187. cos−1 (cos2π

3) 188. cos−1 (cos −

2π

3)

𝟏𝟐

𝟏𝟑

𝟓√𝟔𝟏

𝟔𝟏

𝟑√𝟕

𝟕

−𝟕√𝟐𝟏𝟖

𝟐𝟏𝟖

√𝟖𝟓

𝟏𝟏

−𝟒

𝟑

𝝅

𝟒

𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅

𝝅

𝟑


𝟓

𝟏𝟑

𝟓√𝟕𝟒

𝟕𝟒

√𝟏𝟓

𝟏𝟓

−𝟓√𝟏𝟗𝟒

𝟏𝟗𝟒

𝟐√𝟏𝟎

𝟏𝟏

−𝟑

𝟒

𝝅

𝟔


𝟐𝝅

𝟑



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. 𝑠𝑒𝑐2𝑥 − 2 = 0 192. 2𝑠𝑖𝑛2𝑥 + 3 = 7 sin 𝑥

193. 𝑐𝑠𝑐2𝑥 = 4 194. 3𝑠𝑒𝑐2𝑥 = 4

195. 𝑠𝑖𝑛2𝑥 − cos 𝑥 sin 𝑥 = 0 196. 2(sin 𝑥 + 1) = 𝑐𝑜𝑠2𝑥

197. sin 2𝑥 + cos 𝑥 = 0 198. sin𝑥

2+ cos 𝑥 = 0

199. cos 2𝑥 + cos 𝑥 = 2

𝒙 =𝝅

𝟐

𝒙 =𝝅

𝟔,

𝟓𝝅

𝟔,

𝟕𝝅

𝟔,

𝟏𝟏𝝅

𝟔

𝒙 =𝝅

𝟒,

𝟑𝝅

𝟒,

𝟓𝝅

𝟒,

𝟕𝝅

𝟒

𝒙 =𝝅

𝟔,

𝟓𝝅

𝟔

𝒙 =𝝅

𝟔,

𝟓𝝅

𝟔,

𝟕𝝅

𝟔,

𝟏𝟏𝝅

𝟔

𝒙 =𝝅

𝟔,

𝟓𝝅

𝟔,

𝟕𝝅

𝟔,

𝟏𝟏𝝅

𝟔

𝒙 = 𝟎, 𝝅,𝝅

𝟒,

𝟓𝝅

𝟒

𝒙 =𝟑𝝅

𝟐

𝒙 =𝝅

𝟐,

𝟑𝝅

𝟐,

𝟕𝝅

𝟔,

𝟏𝟏𝝅

𝟔

𝒙 = 𝝅,𝝅

𝟑,

𝟓𝝅

𝟑

𝒙 = 𝟎


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. 𝑐𝑠𝑐2𝑥 − 2 = 0 203. 2𝑠𝑖𝑛2𝑥 − 3 = sin 𝑥

204. 𝑠𝑒𝑐2𝑥 = 4 205. 3𝑐𝑠𝑐2𝑥 = 4

206. 𝑐𝑜𝑠2𝑥 − cos 𝑥 sin 𝑥 = 0 207. (sin 𝑥 − 1) = −2𝑐𝑜𝑠2𝑥

208. sin 2𝑥 = 2tan 2𝑥 209. tan𝑥

2− sin 𝑥 = 0

210. sin 2𝑥 − sin 𝑥 = 0

𝒙 = 𝝅

𝒙 =𝝅

𝟒,

𝟑𝝅

𝟒,

𝟓𝝅

𝟒,

𝟕𝝅

𝟒

𝒙 =𝝅

𝟒,

𝟑𝝅

𝟒,

𝟓𝝅

𝟒,

𝟕𝝅

𝟒

𝒙 =𝟑𝝅

𝟐

𝒙 =𝝅

𝟑,

𝟐𝝅

𝟑,

𝟒𝝅

𝟑,

𝟓𝝅

𝟑

𝒙 =𝝅

𝟑,

𝟐𝝅

𝟑,

𝟒𝝅

𝟑,

𝟓𝝅

𝟑

𝒙 =𝝅

𝟐,

𝟑𝝅

𝟐,

𝝅

𝟒,

𝟓𝝅

𝟒

𝒙 =𝟕𝝅

𝟔,

𝟏𝟏𝝅

𝟔,

𝝅

𝟐

𝒙 = 𝟎, 𝝅,𝝅

𝟐,

𝟑𝝅

𝟐

𝒙 = 𝟎, 𝝅,𝝅

𝟐,

𝟑𝝅

𝟐

𝒙 = 𝟎, 𝝅,𝝅

𝟑,

𝟓𝝅

𝟑


Trigonometry Unit Review

Multiple Choice

1. Given the terminal point of (√2

2,

−√2

2) find tan 𝜃.

a. π

4

b. −π

4

c. -1

d. 1

2. Knowing sec 𝑥 =−5

4 and the terminal point is in the second quadrant find cot 𝜃.

a. −4

5

b. 3

5

c. −4

3

d. −3

4

3. What is the phase shift of 𝑦 =5

3cos(6𝑥 − 2𝜋) + 3?

a. 1

2π

b. π

3

c. 1

3

d. 2𝜋

4. The difference between the maximum of 𝑦 = 2 cos (2 (𝑥 +𝜋

3)) + 1 and 𝑦 = −3 cos(4𝑥 − 𝜋) − 2 is

a. 1

b. 2

c. 3

d. 8

5. 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

6. Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 50°, 𝑎 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.

a. 1.021

b. 40

c. 128.979

d. no solution

7. Given ∆𝐴𝐵𝐶, 𝑤𝑖𝑡ℎ 𝐴 = 50°, 𝑏 = 6, & 𝑐 = 8, 𝑓𝑖𝑛𝑑 𝐵.

a. 6.188

b. 32.456

c. 47.967

d. 82.033

8. (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

𝑐𝑜𝑠2𝑥

d. 1

C

C

B

A

C

D

C

D


9. Find the exact value of sin𝜋

12

a. √6−√2

4

b. √6+√2

4

c. √6−√2

2

d. √6−√2

2

10. On the interval [0, 2π), sin 2𝑥 = 0, thus x =

a. 0

b. π

2

c. 3π

2

d. all of the above

11. Find the exact value of cos 105

a. √2−√3

2

b. −√2−√3

2

c. √2+√3

2

d. −√2+√3

2

12. 𝑠𝑖𝑛4𝑥 =

a. 1

8(3 − cos 𝑥 + cos 4𝑥)

b. 1

8(3 + cos 𝑥 + cos 4𝑥)

c. 1

8(3 + 4 cos 𝑥 + cos 4𝑥)

d. 1

8(3 − 4cos 𝑥 + cos 4𝑥)

13. Rewrite cos 6𝑥 sin 4𝑥 as a sum or difference.

a. 1

2cos 10x −

1

2cos2x

b. 1

2cos 10x +

1

2cos2x

c. 1

2sin 10x − sin2x

d. 1

2sin 10x −

1

2sin2x

14. On the interval [0, 2π), sin 5𝑥 + sin 3𝑥 = 0

a. π

4

b. kπ

4, where k ∈ Integers

c. kπ

4, where k ∈ {0,1,2,6}

d. no solution on the interval given

15. 𝑠𝑖𝑛−1 (sin4𝜋

3) =

a. 4𝜋

3

b. −𝜋

3

c. 𝑏𝑜𝑡ℎ 𝑎 𝑎𝑛𝑑 𝑏

d. Undefined

A

D

B

D

D

C

B


16. On the interval [0, 2π), solve 2sin2 𝑥 + 3 cos 𝑥 = 3

I. 0 II. π

3 III.

5π

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 𝑣0 and angle 𝜃, is

𝑟 =1

16𝑣0

2 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?

D

𝒓 =𝟏

𝟑𝟐𝒗𝟎

𝟐 𝐬𝐢𝐧 𝟐𝜽

𝜽 = 𝟏𝟒. 𝟑𝟒𝒐

𝒓 = 𝟏𝟐𝟓𝟎 𝒇𝒆𝒆𝒕

𝟑𝟐𝟎. 𝟐 𝒇𝒆𝒆𝒕

𝟐𝟏𝟏. 𝟖 𝒇𝒆𝒆𝒕

𝟑𝟕𝟖. 𝟖 𝒇𝒆𝒆𝒕


3. The average daily production, M (in hundreds of gallons), on a dairy farm is modeled by

𝑀 = 19.6 sin (2𝜋𝑑

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 𝑥 + 1 = sin 𝑥

cos2x + 2 cos x + 1 = 𝑠𝑖𝑛2𝑥

cos2x + 2 cos x + 1 = 1 − 𝑐𝑜𝑠2𝑥

2 cos 𝑥 = 0

cos 𝑥 = 0

π

2,3π

2

𝟑𝟔𝟓

𝟒𝟓𝟔𝟔 𝒈𝒂𝒍𝒍𝒐𝒏𝒔

February thru May

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

𝒙 =𝝅

𝟐, 𝝅,

𝟑𝝅

𝟐

Documents

Unit Circle Class Work - Center For Teaching & Learningcontent.njctl.org/courses/math/pre-calculus/trigonometry-2/trigonometry-cw-hw... · Pre-Calc Trig ~1~ NJCTL.org Unit Circle