final exam pre-calculus - somerset...

Final Exam Pre-Calculus

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. If , find .

a. 8

5

c. 6

7

b. 13

5

d. 7

6

____ 2. Find the values of the six trigonometric functions for angle , when and .

a.

sin = 5

3, cos =

5

4, csc =

3

5, sec =

4

5, tan =

3

4, and cot =

4

3.

b. sin =

4

5, cos =

3

5, csc =

5

4, sec =

5

3, tan =

4

3, and cot =

3

4.

c. sin =

5

3, cos =

3

4, csc =

3

5, sec =

4

5, tan =

4

3, and cot =

5

4.

d. sin =

4

5, cos =

3

5, csc =

5

3, sec =

5

4, tan =

4

3, and cot =

4

3.

____ 3. Find the values of the six trigonometric functions for angle , when and .

a.

sin = 12

13, cos =

5

13, csc =

13

12, sec =

13

5, tan =

12

5, and cot =

5

12.

b. sin =

12

13, cos =

5

13, csc =

13

5, sec =

13

12, tan =

12

5, and cot =

12

5.

c. sin =

13

5, cos =

5

12, csc =

5

13, sec =

12

13, tan =

12

5, and cot =

13

12.

d. sin =

13

5, cos =

13

12, csc =

5

13, sec =

12

13, tan =

5

12, and cot =

12

5.

____ 4. If and find . Round to the nearest tenth.

a. c. b. d.

____ 5. If and , find . Round to the nearest tenth.

a. c. b. d.

____ 6. Solve by using the measurements , , and . Round measures of sides

to the nearest tenth and measures of angles to the nearest degree.

A

B Ca

bc

a. , , c. , ,

b. , , d. , ,

____ 7. Solve by using the measurements , , and . Round measures of

sides to the nearest tenth and measures of angles to the nearest degree.

P

Q Rp

qr

a. , , c. , ,

b. , , d. , ,

____ 8. Change 3.94 radians to degree measure. Round to the nearest tenth.

a. c. b. d.

____ 9. Change to radian measure in terms of .

a. 35

54

c. 35

36

b. 35

18

d. 35

72

____ 10. Write in degrees.

a. c. b. d.

____ 11. Write in degrees

a. c. b. d.

____ 12. Write –2160° in radians.

a. c.

b. d.

____ 13. Find one positive and one negative angle coterminal with an angle of 126°.

a. 486°, –234° c. 486°, –36

b. 526°, –54° d. 216°, –36°

____ 14. Find one positive and one negative angle coterminal with an angle of 106°.

a. 466°, –16 c. 474°, –74°

b. 466°, –254° d. 196°, –16°

____ 15. Find the least positive angle measurement that is coterminal with °.

a. ° c. °

b. ° d. °

____ 16. Suppose is an angle in the standard position whose terminal side is in Quadrant III and . Find

the exact values of the five remaining trigonometric functions of .

a. 17

15,

17

8,

15

17,

8

15, and

15

8

c. 17

8,

17

15,

8

17,

8

15, and

15

8

b. –8

17, –

15

17, –

17

15,

15

8, and

8

15

d. 15

17, –

8

17, –

17

15,

15

8, and

8

15

____ 17. Suppose is an angle in the standard position whose terminal side is in Quadrant IV and . Find

the exact values of the five remaining trigonometric functions of

a.

, , , ,

b. , , , ,

c. , , , ,

d. , , , ,

____ 18. Find the values of the six trigonometric functions of an angle in standard position if the point with coordinates

(10, 24) lies on its terminal side.

a. sin =

12

5, cos =

12

13, tan =

5

13

csc = 5

12, sec =

13

12, cot =

13

5

c. sin =

12

13, cos =

5

13, tan =

12

5

csc = 13

12, sec =

13

5, cot =

5

12

b. sin =

5

13, cos =

12

13, tan =

5

12

csc = 13

5, sec =

13

12, cot =

12

5

d. sin =

13

12, cos =

13

5, tan =

5

12

csc = 12

13, sec =

5

13, cot =

12

5

____ 19. Find the values of the six trigonometric functions of an angle in standard position if the point with coordinates

(5, 12) lies on its terminal side.

a. sin =

13

12, cos =

13

5, tan =

5

12

csc = 12

13, sec =

5

13, cot =

12

5

c. sin =

5

13, cos =

12

13, tan =

5

12

csc = 13

5, sec =

13

12, cot =

12

5

b. sin =

12

5, cos =

12

13, tan =

5

13

csc = 5

12, sec =

13

12, cot =

13

5

d. sin =

12

13, cos =

5

13, tan =

12

5

csc = 13

12, sec =

13

5, cot =

5

12

____ 20. Use the unit circle to find the value of .

a. c. b. undefined d.

____ 21. Find the exact value of .

a.

c.

b.

d.


a.

c.

b. d.

____ 23.

a. 2 c.

b.

d.

____ 24.

a.

c.

b.

d. 2

____ 25.

a.

c.

b.

d.


a. 1 c.

b.

d. 0

____ 27. Find the reference angle for 288

a. 89 c. 72

b. 59 d. 108

____ 28. Find the exact value of sin .

a.

c. undefined

b.

d.

____ 29. Use the unit circle to find the value of .

a. c. b. d. undefined

____ 30.

a.

c.

b. 2 d.

____ 31. Find the amplitude, period, and phase shift of

a. amplitude = –5

period =

phase shift = –8

7

b. amplitude = 10

period =

phase shift = 8

7

c. amplitude = –5

period =

phase shift = –8

7

d. amplitude = 5

period =

phase shift = 8

7

____ 32. Write an equation of the cosine function with amplitude 2 and period

a.

c.

b.

d.

____ 33. Find the amplitude of . Then graph the function.

a. amplitude:

90 180 270–90–180–270

0.5

1

1.5

2

2.5

–0.5

–1

–1.5

–2

–2.5

y

c. amplitude: does not exist

90 180 270–90–180–270

0.1

0.2

0.3

0.4

0.5

–0.1

–0.2

–0.3

–0.4

–0.5

y

b. amplitude:

90 180 270–90–180–270

0.1

0.2

0.3

0.4

0.5

–0.1

–0.2

–0.3

–0.4

–0.5

y

d. amplitude:

90 180 270–90–180–270

0.1

0.2

0.3

0.4

0.5

–0.1

–0.2

–0.3

–0.4

–0.5

y

____ 34. Find the amplitude of . Then graph the function.

a. amplitude: 9 c. amplitude: does not exist

90 180 270–90–180–270

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

90 180 270–90–180–270

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

b. amplitude: 9

90 180 270–90–180–270

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

d. amplitude: 1

90 180 270–90–180–270

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

____ 35. Given a triangle with a = , A = °, and B = °, what is the length of c? Round to the nearest tenth.

a. c. b. d.

____ 36. Given a triangle with b = , c = , and A = ° what is the length of a? Round to the nearest tenth.

a. c. b. d.

____ 37. Solve .

A

B C

, ,

a. , , c. , ,

b. , , d. , ,

____ 38. Determine whether should be solved by using the Law of Sines or the Law of Cosines. Then solve the

triangle.

A

B C

c b

a

, ,

a. Law of Sines; , ,

b. Law of Cosines; , ,

c. Law of Cosines; , ,

d. Law of Sines; , ,

____ 39. Determine whether should be solved by using the Law of Sines or the Law of Cosines. Then solve the

triangle.

A

B C

c b

a

, ,

a. Law of Cosines; , ,

b. Law of Cosines; , ,

c. Law of Sines; , ,

d. Law of Sines; , ,

____ 40. If and , find and

a. ,

c. ,

b. ,

d. ,

____ 41. If sin = 3

7, find csc .

a. –3 c.

3

7

b.

7

3

d. 7

____ 42. Simplify cos x – sin x cot x.

a. c. b. d.

____ 43. Simplify .

a. c. b. d.

____ 44. What basic trigonometric identity would you use to verify that tan x cos x = sin x?

a.

c.

b.

d.

____ 45. What basic trigonometric identity would you use to verify that cot x sin x = cos x?

a. c.

b.

d.

____ 46. What basic trigonometric identity would you use to verify that ?

a.

c.

b. d.


a. sin x = cos x tan x c.

b. d.


a.

c. sin x = cos x tan x

b. d.

____ 49. Simplify

a. c. 0

b. d.

Final Exam Pre-Calculus

Answer Section

MULTIPLE CHOICE

1. ANS: B PTS: 1 DIF: Average REF: Lesson 4-1

OBJ: 4-1.1 Find values of trigonometric functions for acute angles of right triangles.

NAT: 3 STA: MA.912.T.1.3 | MA.912.T.2.1 TOP: Right Triangle Trigonometry

KEY: Trigonometry | Trigonometric Ratios

NOT: Example 2: Use One Trigonometric Value to Find Others




KEY: Trigonometric Functions | Acute Angles

NOT: Example 1: Find Values of Trigonometric Ratios

3. ANS: A PTS: 1 DIF: Average REF: Lesson 4-1



KEY: Trigonometric Functions | Acute Angles

NOT: Example 1: Find Values of Trigonometric Ratios

4. ANS: D PTS: 1 DIF: Average REF: Lesson 4-1

OBJ: 4-1.2 Solve right triangles. NAT: 3 STA: MA.912.T.1.4

TOP: Right Triangle Trigonometry KEY: Trigonometry | Trigonometric Ratios | Right Triangles

NOT: Example 3: Find a Missing Side Length



TOP: Right Triangle Trigonometry KEY: Find Angle Measurements

NOT: Example 5: Find a Missing Angle Measure

6. ANS: D PTS: 1 DIF: Advanced REF: Lesson 4-1


TOP: Right Triangle Trigonometry KEY: Solve Triangles | Right Triangles

NOT: Example 8: Solve a Right Triangle

7. ANS: A PTS: 1 DIF: Advanced REF: Lesson 4-1


TOP: Right Triangle Trigonometry KEY: Solve Triangles | Right Triangles

NOT: Example 8: Solve a Right Triangle

8. ANS: C PTS: 1 DIF: Average REF: Lesson 4-2

OBJ: 4-2.1 Convert degree measures of angles to radian measures, and vice versa.

NAT: 3 STA: MA.912.T.1.1 TOP: Degrees and Radians

KEY: Angle Measures | Degree Measures | Radian Measures

NOT: Example 2: Convert Between Degree and Radian Measures




KEY: Angle Measures | Degree Measures | Radian Measures


10. ANS: C PTS: 1 DIF: Basic REF: Lesson 4-2



KEY: Radian Measure | Degree Measures


11. ANS: B PTS: 1 DIF: Basic REF: Lesson 4-2










13. ANS: A PTS: 1 DIF: Basic REF: Lesson 4-2



KEY: Coterminal Angles NOT: Example 3: Find and Draw Coterminal Angles






OBJ: 4-2.2 Use angle measure to solve real-world problems. NAT: 3

STA: MA.912.T.3.4 TOP: Degrees and Radians


16. ANS: D

Feedback

A These are the inverse values. B Ensure that the correct trigonometric functions were used when calculating. C Ensure that the correct trigonometric functions were used when calculating. D Correct!

PTS: 1 DIF: Average REF: Lesson 4-3

OBJ: 4-3.1 Find values of trigonometric functions for any angle.

NAT: 3 STA: MA.912.T.2.1

TOP: Trigonometric Functions on the Unit Circle

KEY: Trigonometric Functions | Trigonometric Ratios

NOT: Example 2: Evaluate Trigonometric Functions of Quadrantal Angles

17. ANS: B

Feedback

A Check the ratios again. Some parts are correct and some are not. B Correct! C Check the ratios again. Some parts are correct and some are not. D These are the inverses of the correct answers.



NAT: 3 STA: MA.912.T.2.1




18. ANS: C

Feedback

A Please review the trigonometric ratios. B There is confusion between the sine and cosine values. C Correct! D These are the inverses of the correct answers.



NAT: 3 STA: MA.912.T.2.1


KEY: Trigonometry | Trigonometric Functions | Unit Circle

NOT: Example 7: Find Trigonometric Values Using the Unit Circle



NAT: 3 STA: MA.912.T.2.1





OBJ: 4-3.2 Find values of trigonometric functions using the unit circle.

NAT: 3 STA: MA.912.T.1.2






NAT: 3 STA: MA.912.T.1.2


KEY: Circular Functions | Unit Circle | Periodic Functions

NOT: Example 8: Use the Periodic Nature of Circular Functions



NAT: 3 STA: MA.912.T.1.2






NAT: 3 STA: MA.912.T.1.2






NAT: 3 STA: MA.912.T.1.2






NAT: 3 STA: MA.912.T.1.2




26. ANS: B

[Change angle measure to one that is on the unit circle]

PTS: 1 DIF: Basic REF: Lesson 4-3


NAT: 3 STA: MA.912.T.1.2


KEY: Trigonometric Functions | Unit Circle


27. ANS: C

Feedback

A This is too large to be the reference angle. B This is too small to be the reference angle. C Correct! D The reference angle refers to 360 degrees, not 180 degrees.



NAT: 3 STA: MA.912.T.1.2


KEY: Trigonometric Functions | Trigonometric Ratios NOT: Example 3: Find Reference Angles

28. ANS: B

Feedback

A Close, the correct answer has the opposite sign. B Correct! C This value is defined. Check your calculations again. D Check the calculations again. This is incorrect.



NAT: 3 STA: MA.912.T.1.2






NAT: 3 STA: MA.912.T.1.2






NAT: 3 STA: MA.912.T.1.2





OBJ: 4-4.1 Graph transformations of the sine and cosine functions.

NAT: 2 STA: MA.912.A.2.10 | MA.912.T.1.6

TOP: Graphing Sine and Cosine Functions

KEY: Amplitude | Period | Sine Functions | Cosine Functions

NOT: Example 5: Graph Horizontal Translations of Sinusoidal Functions



NAT: 2 STA: MA.912.A.2.10 | MA.912.T.1.6

TOP: Graphing Sine and Cosine Functions

KEY: Sine Functions | Cosine Functions | Amplitude | Period

NOT: Example 3: Graph Horizontal Dilations of Sinusoidal Functions

33. ANS: D PTS: 1 DIF: Basic REF: Lesson 4-4


NAT: 2 STA: MA.912.A.2.10 | MA.912.T.1.6

TOP: Graphing Sine and Cosine Functions KEY: Amplitude | Sine | Cosine

NOT: Example 1: Graph Vertical Dilations of Sinusoidal Functions



NAT: 2 STA: MA.912.A.2.10 | MA.912.T.1.6

TOP: Graphing Sine and Cosine Functions KEY: Amplitude | Sine | Cosine

NOT: Example 1: Graph Vertical Dilations of Sinusoidal Functions


OBJ: 4-7.1 Solve oblique triangles by using the Law of Sines ore the Law of Cosines.

NAT: 3 STA: MA.912.T.2.3

TOP: The Law of Sines and the Law of Cosines KEY: Solve Triangles | Law of Sines

NOT: Example 1: Apply the Law of Sines (AAS)



NAT: 3 STA: MA.912.T.2.3

TOP: The Law of Sines and the Law of Cosines KEY: Solve Triangles | Law of Cosines

NOT: Example 6: Apply the Law of Cosines (SAS)



NAT: 3 STA: MA.912.T.2.3

TOP: The Law of Sines and the Law of Cosines KEY: Solve Triangles | Law of Sines

NOT: Example 1: Apply the Law of Sines (AAS)



NAT: 3 STA: MA.912.T.2.3


NOT: Example 5: Apply the Law of Cosines (SSS)



NAT: 3 STA: MA.912.T.2.3


NOT: Example 5: Apply the Law of Cosines (SSS)


OBJ: 5-1.1 Identify and use basic trigonometric identities to find trigonometric values.

NAT: 3 STA: MA.912.T.3.2 TOP: Trigonometric Identities

KEY: Reciprocal Identities | Quotient Identities | Pythagorean Identities | Symmetry Identities |

Opposite-Angle Identities NOT: Example 2: Use Pythagorean Identities

41. ANS: B

Feedback

A The answer should be the reciprocal of the original value. B Correct! C The answer should be the reciprocal of the original value. D The answer should be the reciprocal of the original value.


OBJ: 5-1.1 Identify and use basic trigonometric identities to find trigonometric values.


KEY: Reciprocal Identities | Trigonometric Identities

NOT: Example 1: Use Reciprocal and Quotient Identities


OBJ: 5-1.2 Use basic trigonometric identities to simplify and rewrite trigonometric expressions.


KEY: Trigonometric Identities | Simplify Trigonometric Expressions

NOT: Example 4: Simplify by Rewriting Using Only Sine and Cosine

43. ANS: A

Feedback

A Correct! B Watch your signs. C Watch your basic trig identities. D Watch your signs and identities.


OBJ: 5-2.1 Verify trigonometric identities. NAT: 3

STA: MA.912.T.3.2 TOP: Verifying Trigonmetric Identities

KEY: Verify Trigonometric Identities

NOT: Example 2: Verify a Trigonometric Identity by Combining Fractions




KEY: Trigonometric Identities | Verify Identities

NOT: Example 1: Verify a Trigonometric Identity





















49. ANS: A

Feedback

A Correct! B Watch your signs. C Watch your identities. D Check your multiplication.


OBJ: 5-2.2 Determine whether equations are identities. NAT: 3


KEY: Trigonometric Identities

NOT: Example 3: Verify a Trigonometric Identity by Multiplying

final exam pre-calculus - somerset...

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