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) $\frac { \sqrt { 3 } } { 2 }$

Accepted Answer

Using the identity $\cos \theta = \sin \left( \frac { \pi } { 2 } - \theta \right)$, we can rewrite $\cos \frac { 3 \pi } { 8 }$ as $\sin \frac { \pi } { 8 }$. Thus, $\cos \frac { 3 \pi } { 8 } - \sin \frac { \pi } { 8 } = \sin \frac { \pi } { 8 } - \sin \frac { \pi } { 8 } = 0$.

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) $\frac { \sqrt { 3 } } { 2 }$

Accepted Answer

B

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

Accepted Answer

The product-to-sum formula for cosine is $\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]$. Applying this to $\cos \frac{\pi}{5} \cos \frac{4\pi}{5}$, we get $\frac{1}{2}[\cos(\frac{\pi}{5} - \frac{4\pi}{5}) + \cos(\frac{\pi}{5} + \frac{4\pi}{5})] = \frac{1}{2}[\cos(-\frac{3\pi}{5}) + \cos(\frac{5\pi}{5})] = \frac{1}{2}[\cos(\frac{3\pi}{5}) - 1]$, since $\cos(-\theta) = \cos(\theta)$ and $\cos(\pi) = -1$.

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) $\frac { \sqrt { 26 } } { 26 }$

Accepted Answer

The answer of Refer to the triangle and compute \(\tan...

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

Accepted Answer

Using the identities $\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta$ and $\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta$, and calculating $\cos\alpha = -\frac{5}{13}$ and $\sin\beta = -\frac{8}{17}$, we find $\sin(\alpha - \beta) = -\frac{220}{221}$ and $\cos(\alpha - \beta) = -\frac{21}{221}$.

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

Accepted Answer

The answer of Use the given information to compute \(\cos...

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) $\frac { \sqrt { 77 } } { 15 }$

Accepted Answer

The answer of Use the given information to compute \(\cos...

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.

Accepted Answer

The answer of Determine all of the solutions in the...

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

Accepted Answer

The answer of Find the right side of the identity....

Question 10

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

Accepted Answer

The answer of Is \(t = \frac { \pi }...

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) $- \frac { 119 } { 120 }$

Accepted Answer

Using the sine formula, we have:

$$\sin2s=2\sin s \cos s$$

Applying the Pythagorean theorem in the triangle, we have $\cos s = \frac{120}{169}$ and $\sin s = \frac{119}{169}$.

Substituting in the formula we have:

$$\sin2s=2\cdot \frac{119}{169}\cdot \frac{120}{169}=\frac{28560}{28561}=\frac{120^2}{120^2+119^2}$$

Therefore, the answer is A.

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

Accepted Answer

We can use the product-to-sum formula $\cos a \cos b = \frac{1}{2} (\cos(a-b) + \cos(a+b))$. 
Plugging in $a=47^\circ$ and $b=43^\circ$, we get: 
$$\cos 47^\circ \cos 43^\circ = \frac{1}{2}(\cos 4^\circ + \cos 90^\circ) = \frac{1}{2} \cos 4^\circ$$
since $\cos 90^\circ=0$. 
Therefore, the answer is (C), $\frac{1}{2} \cos 4^\circ$.

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

Accepted Answer

The answer of Use the addition formulas for tangent to...

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

Accepted Answer

Using the product-to-sum formula: $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$, we get $\frac{1}{2}[\cos(\frac{6\pi}{12}) + \cos(\frac{4\pi}{12})] = \frac{1}{2}[\cos(\frac{\pi}{2}) + \cos(\frac{\pi}{3})] = \frac{1}{2}[0 + \frac{1}{2}] = \frac{1}{4}$.

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

Accepted Answer

The answer of Use the following information to evaluate the...

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) $\frac { 1 } { 2 } \cos 130 ^ { \circ } - \frac { 1 } { 4 }$

Accepted Answer

Using the product-to-sum formula, we have:

$$\sin 125^\circ \sin 5^\circ = \frac{1}{2}\left(\cos(125^\circ-5^\circ)-\cos(125^\circ+5^\circ)\right)$$

Simplifying, we get:

$$\sin 125^\circ \sin 5^\circ = -\frac{1}{4}-\frac{1}{2}\cos 130^\circ$$

So the answer is choice B.

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 $\cos \theta$

Accepted Answer

The answer of Use the addition formulas for sine and...

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

Accepted Answer

We can use the formulas for the tangent of the sum and difference of two angles: 
$$	an (s+t)=\frac{	an s+	an t}{1-	an s	an t}$$
$$	an (s-t)=\frac{	an s-	an t}{1+	an s	an t}$$
Substituting the given values, we have:
$$	an (s+t)=\frac{5+4}{1-(5)(4)}=-\frac{9}{19}$$
$$	an (s-t)=\frac{5-4}{1+(5)(4)}=\frac{1}{21}$$
Therefore, the answer is A.

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) $\frac { \sqrt { 3 } } { 4 }$

Accepted Answer

The answer of Use the formula for \(\sin ( s...

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

Accepted Answer

The answer of Compute $\cos ( \alpha + \theta )$...

Quiz 9: Analytical Trigonometry

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

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

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 $\tan \left( \frac { t } { 2 } \right)$ . $Refer to the triangle and compute \tan \left( \frac { t } { 2 } \right) . A) 1/5 B) \frac { 5 \sqrt { 13 } } { 26 } C) \frac { \sqrt { 6 } } { 26 } D) 1/8 E) \frac { \sqrt { 26 } } { 26 }$

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

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

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. $\frac { \sin s + \sin t } { \sin s + \sin t } = ?$

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

Refer to the triangle and compute $\sin 2 s$ . $Refer to the triangle and compute \sin 2 s . A) \frac { 120 } { 169 } B) - \frac { 245 } { 389 } C) \frac { 201 } { 389 } D) - \frac { 119 } { 169 } E) - \frac { 119 } { 120 }$

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

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

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

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

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

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

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

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

Quiz 9: Analytical Trigonometry

Use the addition formulas for sine and cosine to simplify the expression. sin⁡54∘cos⁡24∘−cos⁡54∘sin⁡24∘\sin 54 ^ { \circ } \cos 24 ^ { \circ } - \cos 54 ^ { \circ } \sin 24 ^ { \circ }sin54∘cos24∘−cos54∘sin24∘

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​) .

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 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

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​=?

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 ?

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

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∘

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​​

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. cos⁡5π12cos⁡π12\cos \frac { 5 \pi } { 12 } \cos \frac { \pi } { 12 }cos125π​cos12π​

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​) =?

Use a product-to-sum formula to convert the expression to a sum or difference. Simplify where possible. sin⁡125∘sin⁡5∘\sin 125 ^ { \circ } \sin 5 ^ { \circ }sin125∘sin5∘

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 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

Use the formula for sin⁡(s−t) \sin ( s - t ) sin(s−t) to compute the exact value of sin⁡π12\sin \frac { \pi } { 12 }sin12π​ .

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

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 $\tan \left( \frac { t } { 2 } \right)$ . $Refer to the triangle and compute \tan \left( \frac { t } { 2 } \right) . A) 1/5 B) \frac { 5 \sqrt { 13 } } { 26 } C) \frac { \sqrt { 6 } } { 26 } D) 1/8 E) \frac { \sqrt { 26 } } { 26 }$

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

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. $\frac { \sin s + \sin t } { \sin s + \sin t } = ?$

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

Refer to the triangle and compute $\sin 2 s$ . $Refer to the triangle and compute \sin 2 s . A) \frac { 120 } { 169 } B) - \frac { 245 } { 389 } C) \frac { 201 } { 389 } D) - \frac { 119 } { 169 } E) - \frac { 119 } { 120 }$

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

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

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

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

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

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

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