Deck 6: Analytic Trigonometry

Question 1

Find all solutions of the equation. $$( \cos \theta - 1 ) ( \sin \theta + 1 ) = 0$$

A) $2 \pi$ and $- \frac { \pi } { 2 } + 2 \pi n$ for every integer n.
B) $2 \pi$ and $\pi + 2 \pi n$ for every integer n.
C) $\pi + 2 \pi n$ and $\frac { \pi } { 2 } + 2 \pi n$ for every integer n.
D) $\frac { \mathrm { mn } } { 2 }$ for every integer n.
E) $\pi + 2 \pi n$ and $- \frac { \pi } { 2 } + \pi n$ for every integer n.

Answer 1

$2 \pi$ and $- \frac { \pi } { 2 } + 2 \pi n$ for every integer n.

Question 2

If and for a second-quadrant angle and a third-quadrant angle , find

A)
B)
C)
D)
E)

Answer 2

C

Question 3

Find the solutions of the equation that are in the interval

A)
B)
C)
D)
E) no solution

Answer 3

D

Question 4

Express as a product.

A)
B)
C)
D)
E)

Question 5

Verify the identity.

A) invalid
B) valid

Question 6

Use an addition or subtraction formula to find the solutions of the equation that are in the interval $[ 0 , \pi )$ . $\cos 8 t \cos 5 t = - \sin 8 t \sin 5 t$

A) $t = \frac { \pi } { 6 } , t = \frac { \pi } { 2 } , t = \frac { 5 \pi } { 6 }$
B) $t = \frac { 3 \pi } { 5 } , t = \frac { 4 \pi } { 5 }$
C) $t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$
D) $t = \frac { \pi } { 8 } , t = \frac { 3 \pi } { 8 } , t = \frac { 5 \pi } { 8 }$
E) $t = 0 , t = \frac { \pi } { 3 } , t = \frac { 2 \pi } { 3 }$

Question 7

Verify the identity.

A) Not identical
B) Identical

Question 8

Express as a sum or difference.

A)
B)
C)
D)
E)

Question 9

Find the exact values of for the given values of .

A)
B)
C)
D)
E)

Question 10

Use the graph of f to find the simplest expression g(x) such that the equation is an identity.

A)
B)
C)
D)
E)

Question 11

Make the trigonometric substitution and a > 0. Use fundamental identities to simplify the resulting expression.

A)
B)
C)
D)
E)

Question 12

Verify the identity. $$\sin ^ { 3 } t + \cos ^ { 3 } t = ( 1 - \sin t \cos t ) ( \sin t + \cos t )$$

A) valid
B) invalid

Question 13

Verify the identity.

A) valid
B) invalid

Question 14

Express as a product. $\sin 13 \theta + \sin 3 \theta$

A) $\sin ( 8 \theta ) \cos ( 5 \theta )$
B) $2 \sin ( 16 \theta ) \cos ( 10 \theta )$
C) $2 \cos ( 9 \theta ) \sin ( 4 \theta )$
D) $4 \sin ( 8 \theta ) \cos ( 5 \theta )$
E) $2 \sin ( 8 \theta ) \cos ( 5 \theta )$

Question 15

Use the formula to determine the amplitude of f (x).

A)
B)
C)
D)
E)

Question 16

Verify the identity.

A) Not identical
B) Identical

Question 17

Verify the identity.

A) valid
B) invalid

Question 18

Verify the identity. $$\ln \sec \theta = - \ln \cos \theta$$

A) valid
B) invalid

Question 19

Use the formula to determine the period of f (x).

A)
B)
C)
D)
E)

Question 20

Find all solutions of the equation.

A) for every integer n.
B) for every integer n.
C) for every integer n.
D) for every integer n.
E) for every integer n.

Question 21

Find an identical expression.

A)
B)
C)
D)
E)

Question 22

Verify the identity.

A) Not identical
B) Identical

Question 23

Use the graph of f to find the simplest expression g(x) such that the equation f (x) = g (x) is an identity.

A)
B)
C)
D)
E)

Question 24

A golfer, centered in a b = 40-yard-wide straight fairway, hits a ball a = 230 yards. Approximate the largest angle the drive can have from the center of the fairway if the ball is to stay in the fairway (see the figure). Round the answer to the nearest hundredth.

A)
B)
C)
D)
E)

Question 25

If an earthquake has a total horizontal displacement of S meters along its fault line, then the horizontal movement M of a point on the surface of Earth d kilometers from the fault line can be estimated using the formula where D is the depth (in kilometers) below the surface of the focal point of the earthquake. For the San Francisco earthquake of 1906, S was 4 meters and D was 3.08 kilometers. Approximate M for d = 6 kilometers. Round the answer to the nearest hundredth.

A) m
B) m
C) m
D) m
E) m