Question 1

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

Accepted Answer

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)  $\tan ( s + t ) = - \frac { 21 } { 1 }$ $\tan ( s - t ) = \frac { 19 } { 9 }$ 
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)  $\tan ( s + t ) = - \frac { 21 } { 1 }$ $\tan ( s - t ) = \frac { 19 } { 9 }$ A

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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)  $\frac { 1 } { 2 } \left( 1 - \sin 4 ^ { \circ } \right)$ 
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 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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)  $\cos ( \alpha + \theta ) = - \frac { 171 } { 221 }$ $\cos ( \alpha - \theta ) = \frac { 21 } { 221 }$ 
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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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)  $\frac { \sqrt { 5 - 3 \sqrt { 10 } } } { 3 }$ 
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 13

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

Accepted Answer

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)  $\sin ( \alpha - \beta ) = - \frac { 21 } { 221 }$ $\cos ( \alpha - \beta ) = - \frac { 220 } { 221 }$ 
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 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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. 
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 16

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

Accepted Answer

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)  $\tan \left( \frac { s - t } { 2 } \right)$ 
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 17

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

Accepted Answer

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)  $1 - \sin ^ { 2 } \frac { \pi } { 5 }$ 
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)  $1 - \sin ^ { 2 } \frac { \pi } { 5 }$

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

A) True 
 B)False

Question 20

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

Accepted Answer

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Exam 9: Analytical Trigonometry

Use the given information to compute tan⁡(s+t)\tan ( s + t )tan(s+t) and tan⁡(s−t)\tan ( s - t )tan(s−t) . tan⁡s=5\tan s = 5tans=5 and tan⁡t=4\tan t = 4tant=4

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

Use the addition formulas for sine and cosine to simplify the expression. cos⁡(θ−π3)+cos⁡(θ+π3)\cos \left( \theta - \frac { \pi } { 3 } \right) + \cos \left( \theta + \frac { \pi } { 3 } \right)cos(θ−3π​)+cos(θ+3π​)

Use the addition formulas for tangent to simplify the expression. tan⁡2π7−tan⁡5π421+tan⁡2π7tan⁡5π42\frac { \tan \frac { 2 \pi } { 7 } - \tan \frac { 5 \pi } { 42 } } { 1 + \tan \frac { 2 \pi } { 7 } \tan \frac { 5 \pi } { 42 } }1+tan72π​tan425π​tan72π​−tan425π​​

Evaluate the quantity without using a calculator or tables. tan⁡(arccos⁡1213)\tan \left( \arccos \frac { 12 } { 13 } \right)tan(arccos1312​)

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. cos⁡47∘cos⁡43∘\cos 47 ^ { \circ } \cos 43 ^ { \circ }cos47∘cos43∘

Convert the expression into a product. Simplify where possible. cos⁡3π8−sin⁡π8\cos \frac { 3 \pi } { 8 } - \sin \frac { \pi } { 8 }cos83π​−sin8π​ (Hint: Use the identity cos⁡θ=sin⁡(π2−θ)\cos \theta = \sin \left( \frac { \pi } { 2 } - \theta \right)cosθ=sin(2π​−θ) .)

Refer to the triangle and compute sin⁡2s\sin 2 ssin2s .

Evaluate the given quantity, but do not use a calculator or table. cos⁡−1(−12)\cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right)cos−1(−21​)

Evaluate the quantity without using a calculator or tables. cos⁡(arctan⁡13)\cos \left( \arctan \frac { 1 } { \sqrt { 3 } } \right)cos(arctan3​1​)

Use the given information to compute cos⁡(θ2)\cos \left( \frac { \theta } { 2 } \right)cos(2θ​) . sin⁡θ=−18\sin \theta = - \frac { 1 } { 8 }sinθ=−81​ and 180∘<θ<270∘180 ^ { \circ } < \theta < 270 ^ { \circ }180∘<θ<270∘

Use the following information to evaluate the expression. cos⁡θ=1213(3π2<θ<2π)\cos \theta = \frac { 12 } { 13 } \quad \left( \frac { 3 \pi } { 2 } < \theta < 2 \pi \right)cosθ=1312​(23π​<θ<2π) tan⁡(θ2)=?\tan \left( \frac { \theta } { 2 } \right) = ?tan(2θ​)=?

Determine all of the solutions in the interval 0∘≤θ≤180∘0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }0∘≤θ≤180∘ . tan⁡2θ=−33\tan 2 \theta = - \frac { \sqrt { 3 } } { 3 }tan2θ=−33​​

Find the right side of the identity. sin⁡s+sin⁡tsin⁡s+sin⁡t=?\frac { \sin s + \sin t } { \sin s + \sin t } = ?sins+sintsins+sint​=?

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. cos⁡π5cos⁡4π5\cos \frac { \pi } { 5 } \cos \frac { 4 \pi } { 5 }cos5π​cos54π​

Refer to the triangle and compute tan⁡(t2)\tan \left( \frac { t } { 2 } \right)tan(2t​) .

Is t=π3t = \frac { \pi } { 3 }t=3π​ a solution of 2sin⁡t+2cos⁡t=3−12 \sin t + 2 \cos t = \sqrt { 3 } - 12sint+2cost=3​−1 ?

Use the given information to compute cos⁡(θ2)\cos \left( \frac { \theta } { 2 } \right)cos(2θ​) . cos⁡θ=47\cos \theta = \frac { 4 } { 7 }cosθ=74​ and 3π2<θ<2π\frac { 3 \pi } { 2 } < \theta < 2 \pi23π​<θ<2π

Fundamentals

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Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Additional Topics in Algebra

Filters

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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