Spring Break Review - Pre-calculus

Spring Break Review – Basic Trigonometry

Name ________________________________________ Date _____________________________ Period _____

SHOW ALL WORK. WRITE IN PENCIL. Points will be deducted work is not in pencil or if work is not shown for EVERY problem needing work.

Fill in the blank the matches the unit circle to the left. Radians

Point

Degrees

1. _______________

2. _______________

3. _______________

4. _______________

5. _______________

6. _______________

7. _______________

8. _______________

9. _______________

10. _______________

11. _______________

12. ______________

13. _______________

14. _______________

15. ______________

16. What radian measure is equal to 245° ?

18. What degree measure is equal to

3𝜋 4

?

17. What radian measure is equal to 315° ?

19. What degree measure is equal to

11𝜋 6

?

20. What is a coterminal angle between 0 and 360° that is equivalent to 850° ?

21. What is a coterminal angle between 0 and 360° that is equivalent to −900° ?

22. What is a coterminal angle between 48𝜋 0 and 2𝜋 that is equivalent to 6 ?

23. What is a coterminal angle between −39𝜋 0 and 2𝜋 that is equivalent to 6 ?

Find the exact value of each trigonometric function. 24.

25.

26.

27.

Find the reference angle. 28.

29.

Identify the AMPLITUDE on the following function. 30.

𝜋

𝑓(𝑥 ) = −3sin (2𝜃 + ) − 1 2

Identify the PHASE SHIFT on the following function. 32.

1

𝑓(𝑥 ) = − sin(2𝜃 + 2𝜋) + 5 2

Graph the following functions.

34. 𝑓(𝑥 ) = −cos(𝜃 )

𝜋

35. 𝑓(𝑥 ) = sin (𝜃 − ) 2

Identify the VERTICAL SHIFT on the following function. 𝜋 31. 𝑓 (𝑥 ) = −2sin (𝜃 − ) + 6 2

Identify the PERIOD on the following function. 33.

𝜋

𝑓(𝑥 ) = 2sin (𝜃 + ) + 5 2

Solve for the missing part in the triangle. 36.

37.

38.

39.

40. Find the area of the following triangle.

41. Find the area of the following triangle.

2 radar stations that are 9 miles apart located an unidentified plane that vanished from their screens at the same time. The first station indicated that the position of the plane made an angle of 32° with the line between the stations. The second station indicated that it made an angle of 40° with the same line. 42. How far is each station from the point where they lost contact with the plane?

43. A boat leaves Savannah and heads due east for 15 miles. At the same time, a 2nd boat travels in a direction 30° southwest from Savannah for 18 miles. How far apart are the boats?

44. Bobby’s phone can send and receive calls if it is within 30 miles of a transmission tower. On a trip, Bobby drives north past a transmission tower on I-85 for 29 miles, and then he turns at a 88° angle onto Toothless Boulevard and drives for another 26 miles towards Danielsville. Is Robert close enough to the transmission tower to be able to send and receive calls? Defend your answer by using the Law of Cosines.

45. Solve for x. Area = 23 cm2

Simplify the following trigonometric identities.

46. cos 𝑥 tan 𝑥 ∙ csc 𝑥 = 1

47. sin2 𝑥 (1 + cot 2 𝑥) = 1

48. 1 − cot 2 𝑥 sin2 𝑥 = sin2 𝑥

49. (tan2 𝑥 + 1)(1 − sin2 𝑥) = 1

50. (sec 2 𝑥 − 1)(csc 2 𝑥 − 1) = 1

51. cot 2 𝑥 ∙ sin 𝑥 ∙ sec 𝑥 = cot 𝑥

52. (sec 2 𝑥 − 1) ∙

1−sin2 𝑥 1−cos2 𝑥

=1

53.

sin 𝑥 1 − cos2 𝑥

= csc 𝑥

54.

sec 𝑥 1 + tan2 𝑥

= cos 𝑥

56. sec 2 𝑥 (1 − cos 2 𝑥) = tan2 𝑥

58.

60.

cos2 𝑥 sin 𝑥

+ sin 𝑥 = csc 𝑥

cot 𝑥+tan 𝑥 cot 𝑥

= sec 2 𝑥

55.

tan2 𝑥 sec2 𝑥

= sin2 𝑥

57. 1 − cos 2 𝑥 tan2 𝑥 = cos 2 𝑥

59.

61.

1 1+cos 𝑥

+

1 1−cos 𝑥

sec 𝑥−cos 𝑥 sec 𝑥

= 2csc 2 𝑥

= sin2 𝑥

62. Find the determinant of the following matrix:

63. Solve for x and y using Cramer’s Rule.

64. Find the determinant of the following matrix:

65. Solve for x and y using Cramer’s Rule.

66. Find the determinant of the following matrix:

67. Solve for y in the following system:

68. Find the determinant of the following matrix:

69. Solve for y in the following system:

Convert the following equations from General into Standard form. 70.

4𝑥 2 + 9𝑦 2 − 16𝑥 − 36𝑦 − 92 = 0

72. Write the equation of the circle with a center at (8, -3) and a radius of √7 in standard form.

71.

4𝑥 2 − 9𝑦 2 + 40𝑥 − 54𝑦 − 125 = 0

73. Solve the following system of equations: 𝑥 2 + 𝑦 2 = 40 𝑥 = 3𝑦

74. Write the equation of the parabola with focus (6, -3) and directrix 𝑥 = 2.

Equation: _____________________________________

75. Write the equation of the parabola with focus (-4, 1) and vertex (-4, -7).

Equation: _____________________________________ 76. Find the equation of the line tangent to the circle (𝑥 + 2)2 + (𝑦 − 5)2 = 80 at the point (2, −3).

77. What are the co-vertices in the following equation?

(𝑥 − 3)2 (𝑦 + 2)2 + =1 16 25

Vertices: _________________________________

78. What are the Foci in the following equation?

𝑥 2 (𝑦 − 3)2 + =1 25 16

Foci: _________________________________

79. What are the equations of the asymptotes for the following equation?

(𝑦 + 3)2 (𝑥 + 2)2 − =1 4 16

Asymptote #1: _________________________________

80. What are the Foci in the following equation?

𝑦2 64

−

(𝑥−4)2 36

=1

Foci: _________________________________

Asymptote #2: _________________________________