Deck 9: Analytical Trigonometry

Question 1

Convert the expression into a product. Simplify where possible. $\cos \frac { 3 \pi } { 8 } - \sin \frac { \pi } { 8 }$ (Hint: Use the identity $\cos \theta = \sin \left( \frac { \pi } { 2 } - \theta \right)$ .)

A) $\frac { \sqrt { 2 } } { 2 }$
B) 1/2
C) 1
D) 0
E) $\frac { \sqrt { 3 } } { 2 }$

Answer 1

0

Question 2

Use the addition formulas for sine and cosine to simplify the expression. $$\sin 54 ^ { \circ } \cos 24 ^ { \circ } - \cos 54 ^ { \circ } \sin 24 ^ { \circ }$$

A) $\frac { \sqrt { 2 } } { 2 }$
B) 1/2
C) 0
D) 1
E) $\frac { \sqrt { 3 } } { 2 }$

Answer 2

1/2

Question 3

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. $\cos \frac { \pi } { 5 } \cos \frac { 4 \pi } { 5 }$

A) $\frac { 1 } { 2 } \left( \cos \frac { 3 \pi } { 5 } + 1 \right)$
B) $\frac { 1 } { 2 } \left( \cos \frac { 3 \pi } { 5 } - 1 \right)$
C) $1 + \sin ^ { 2 } \frac { \pi } { 5 }$
D) $- \frac { 1 } { 2 } \sin \frac { 3 \pi } { 5 }$
E) $1 - \sin ^ { 2 } \frac { \pi } { 5 }$

Answer 3

$\frac { 1 } { 2 } \left( \cos \frac { 3 \pi } { 5 } - 1 \right)$

Question 4

Refer to the triangle and compute $\tan \left( \frac { t } { 2 } \right)$ .

A) 1/5
B) $\frac { 5 \sqrt { 13 } } { 26 }$
C) $\frac { \sqrt { 6 } } { 26 }$
D) 1/8
E) $\frac { \sqrt { 26 } } { 26 }$

Question 5

Compute $\sin ( \alpha - \beta )$ and $\cos ( \alpha - \beta )$ using the data below. $\sin \alpha = \frac { 12 } { 13 } \quad \text { where } \frac { \pi } { 2 } < \alpha < \pi$ $$\cos \beta = - \frac { 15 } { 17 } \quad \text { where } \pi < \beta < \frac { 3 \pi } { 2 }$$

A) $\sin ( \alpha - \beta ) = \frac { 140 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 171 } { 221 }$
B) $\sin ( \alpha - \beta ) = - \frac { 220 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 21 } { 221 }$
C) $\sin ( \alpha - \beta ) = - \frac { 171 } { 221 }$ $\cos ( \alpha - \beta ) = \frac { 140 } { 221 }$
D) $\sin ( \alpha - \beta ) = \frac { 220 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 21 } { 221 }$
E) $\sin ( \alpha - \beta ) = - \frac { 21 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 220 } { 221 }$

Question 6

Use the given information to compute $\cos \left( \frac { \theta } { 2 } \right)$ . $\sin \theta = - \frac { 1 } { 8 }$ and $180 ^ { \circ } < \theta < 270 ^ { \circ }$

A) $- \frac { 3 \sqrt { 10 } } { 5 }$
B) $\frac { 3 \sqrt { 6 } } { 14 }$
C) $- \frac { \sqrt { 8 - 3 \sqrt { 7 } } } { 4 }$
D) $- \frac { \sqrt { 5 + 3 \sqrt { 7 } } } { 3 }$
E) $\frac { \sqrt { 5 - 3 \sqrt { 10 } } } { 3 }$

Question 7

Use the given information to compute $\cos \left( \frac { \theta } { 2 } \right)$ . $\cos \theta = \frac { 4 } { 7 }$ and $\frac { 3 \pi } { 2 } < \theta < 2 \pi$

A) $- \frac { 14 } { 15 }$
B) $- \frac { \sqrt { 154 } } { 14 }$
C) $\frac { \sqrt { 143 } } { 14 }$
D) 4/15
E) $\frac { \sqrt { 77 } } { 15 }$

Question 8

Determine all of the solutions in the interval $0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }$ . $\tan 2 \theta = - \frac { \sqrt { 3 } } { 3 }$

A) $\theta \approx 75 ^ { \circ }$ , and $\theta \approx 105 ^ { \circ }$
B) $\theta \approx 195 ^ { \circ }$ , and $\theta \approx 165 ^ { \circ }$
C) $\theta \approx 75 ^ { \circ }$ , and $\theta \approx 165 ^ { \circ }$
D) $\theta \approx 15 ^ { \circ }$ , and $\theta \approx 105 ^ { \circ }$
E) $\theta \approx 150 ^ { \circ }$ , and $\theta \approx 330 ^ { \circ }$
F) $$\theta \approx 195 ^ { \circ } \quad, and \quad \theta \approx 285 ^ { \circ }$$

G) no solution.

Question 9

Find the right side of the identity. $\frac { \sin s + \sin t } { \sin s + \sin t } = ?$

A) $\tan \left( \frac { s + t } { 2 } \right)$
B) $\tan ( s - t )$
C) $\sin ( s + t )$
D) $$\cot \left( \frac { s + t } { 2 } \right)$$
E) $\tan \left( \frac { s - t } { 2 } \right)$

Question 10

Is $t = \frac { \pi } { 3 }$ a solution of $2 \sin t + 2 \cos t = \sqrt { 3 } - 1$ ?

Question 11

Refer to the triangle and compute $\sin 2 s$ .

A) $\frac { 120 } { 169 }$
B) $- \frac { 245 } { 389 }$
C) $\frac { 201 } { 389 }$
D) $- \frac { 119 } { 169 }$
E) $- \frac { 119 } { 120 }$

Question 12

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. $\cos 47 ^ { \circ } \cos 43 ^ { \circ }$

A) $\frac { 1 } { 2 } \left( 1 + \sin 4 ^ { \circ } \right)$
B) $\sqrt { 2 } \cos 2 ^ { \circ }$
C) $\frac { 1 } { 2 } \cos 4 ^ { \circ }$
D) $- \frac { 1 } { 2 } \cos 4 ^ { \circ }$
E) $\frac { 1 } { 2 } \left( 1 - \sin 4 ^ { \circ } \right)$

Question 13

Use the addition formulas for tangent to simplify the expression. $\frac { \tan \frac { 2 \pi } { 7 } - \tan \frac { 5 \pi } { 42 } } { 1 + \tan \frac { 2 \pi } { 7 } \tan \frac { 5 \pi } { 42 } }$

A) $\sqrt { 2 }$
B) $\frac { \sqrt { 3 } } { 3 }$
C) $\sqrt { 3 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) 1

Question 14

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. $\cos \frac { 5 \pi } { 12 } \cos \frac { \pi } { 12 }$

A) 0
B) -1/4
C) -1/2
D) 1/4
E) 1/2

Question 15

Use the following information to evaluate the expression. $\cos \theta = \frac { 12 } { 13 } \quad \left( \frac { 3 \pi } { 2 } < \theta < 2 \pi \right)$ $\tan \left( \frac { \theta } { 2 } \right) = ?$

A) $- \frac { \sqrt { 17 } } { 13 }$
B) $\frac { \sqrt { 11 } } { 12 }$
C) 8/7
D) -1/3
E) -1/5

Question 16

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. $\sin 125 ^ { \circ } \sin 5 ^ { \circ }$

A) $\frac { 1 } { 2 } \sin 130 ^ { \circ } - \frac { \sqrt { 3 } } { 4 }$
B) $- \frac { 1 } { 4 } - \frac { 1 } { 2 } \cos 130 ^ { \circ }$
C) $\frac { 1 } { 2 } \sin 130 ^ { \circ } + \frac { \sqrt { 3 } } { 4 }$
D) $\sqrt { 3 } \cos 65 ^ { \circ }$
E) $\frac { 1 } { 2 } \cos 130 ^ { \circ } - \frac { 1 } { 4 }$

Question 17

Use the addition formulas for sine and cosine to simplify the expression. $\cos \left( \theta - \frac { \pi } { 3 } \right) + \cos \left( \theta + \frac { \pi } { 3 } \right)$

A) $\sqrt { 3 }$ $\cos \theta$
B) $\cos \theta$
C) $\frac { 1 } { 3 } \cos \theta$
D) 0
E) 3 $\cos \theta$

Question 18

Use the given information to compute $\tan ( s + t )$ and $\tan ( s - t )$ . $\tan s = 5$ and $\tan t = 4$

A) $\tan ( s + t ) = - \frac { 9 } { 19 }$ $\tan ( s - t ) = \frac { 1 } { 21 }$
B) $\tan ( s + t ) = - \frac { 1 } { 21 }$ $\tan ( s - t ) = \frac { 9 } { 19 }$
C) $\tan ( s + t ) = - \frac { 19 } { 9 }$ $\tan ( s - t ) = \frac { 21 } { 1 }$
D) $\tan ( s + t ) = \frac { 5 } { 6 }$ $\tan ( s - t ) = - \frac { 5 } { 6 }$
E) $\tan ( s + t ) = - \frac { 21 } { 1 }$ $\tan ( s - t ) = \frac { 19 } { 9 }$

Question 19

Use the formula for $\sin ( s - t )$ to compute the exact value of $\sin \frac { \pi } { 12 }$ .

A) $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$
B) $\frac { \sqrt { 3 } - \sqrt { 2 } } { 4 }$
C) $\frac { \sqrt { 3 } + \sqrt { 2 } } { 4 }$
D) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$
E) $\frac { \sqrt { 3 } } { 4 }$

Question 20

Compute $\cos ( \alpha + \theta )$ and $\cos ( \alpha - \theta )$ using the data below. $\sin \alpha = \frac { 3 } { 5 } \quad \text { where } \frac { \pi } { 2 } < \alpha < \pi$ $\cos \theta = \frac { 5 } { 13 } \quad \text { where } - 2 \pi < \theta < - \frac { 3 \pi } { 2 }$

A) $\cos ( \alpha + \theta ) = \frac { 21 } { 221 }$ $\cos ( \alpha - \theta ) = - \frac { 171 } { 221 }$
B) $\cos ( \alpha + \theta ) = - \frac { 56 } { 65 } \cos ( \alpha - \theta ) = \frac { 16 } { 65 }$
C) $\cos ( \alpha + \theta ) = \frac { 16 } { 65 }$ $\cos ( \alpha - \theta ) = - \frac { 56 } { 65 }$
D) $\cos ( \alpha + \theta ) = \frac { 56 } { 65 }$ $\cos ( \alpha - \theta ) = \frac { 16 } { 65 }$
E) $\cos ( \alpha + \theta ) = - \frac { 171 } { 221 }$ $\cos ( \alpha - \theta ) = \frac { 21 } { 221 }$

Question 21

Evaluate the quantity that is defined, but do not use a calculator or tables. $\arcsin \left( - \frac { 1 } { 2 } \right)$

A) -π/4
B) -π/3
C) -π/2
D) π/4
E) -π/6

Question 22

Evaluate the quantity without using a calculator or tables. $\tan \left( \arccos \frac { 12 } { 13 } \right)$

A) $\frac { 12 } { 13 }$
B) $\frac { 5 } { 12 }$
C) $\frac { 5 } { 13 }$
D) $\frac { 13 } { 12 }$
E) $\frac { 13 } { 5 }$

Question 23

Evaluate the given quantity, but do not use a calculator or table. $\cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$

A) π
B) $\frac { \pi } { 2 }$
C) 0
D) $\frac { \pi } { 3 }$
E) $\frac { 2 \pi } { 3 }$

Question 24

Evaluate the quantity without using a calculator or tables. $\sin \left( \tan ^ { - 1 } ( - 1 ) \right)$

A) $0$
B) $- \frac { \sqrt { 2 } } { 2 }$
C) $\frac { \sqrt { 2 } } { 2 }$
D) $\sqrt { 2 }$
E) $- \sqrt { 2 }$

Question 25

Evaluate the quantity without using a calculator or tables. $\cos \left( \arctan \frac { 1 } { \sqrt { 3 } } \right)$

A) 1/2
B) $\frac { \sqrt { 3 } } { 2 }$
C) $- \frac { \sqrt { 3 } } { 2 }$
D) -1/2
E) $\frac { \sqrt { 3 } } { 3 }$