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) =−919	an ( s + t )  = - \frac { 9 } { 19 }tan(s+t) =−199​   tan⁡(s−t) =121	an ( s - t )  = \frac { 1 } { 21 }tan(s−t) =211​ 
B)    tan⁡(s+t) =−121	an ( s + t )  = - \frac { 1 } { 21 }tan(s+t) =−211​   tan⁡(s−t) =919	an ( s - t )  = \frac { 9 } { 19 }tan(s−t) =199​ 
C)    tan⁡(s+t) =−199	an ( s + t )  = - \frac { 19 } { 9 }tan(s+t) =−919​   tan⁡(s−t) =211	an ( s - t )  = \frac { 21 } { 1 }tan(s−t) =121​ 
D)    tan⁡(s+t) =56	an ( s + t )  = \frac { 5 } { 6 }tan(s+t) =65​   tan⁡(s−t) =−56	an ( s - t )  = - \frac { 5 } { 6 }tan(s−t) =−65​ 
E)    tan⁡(s+t) =−211	an ( s + t )  = - \frac { 21 } { 1 }tan(s+t) =−121​   tan⁡(s−t) =199	an ( s - t )  = \frac { 19 } { 9 }tan(s−t) =919​ A

Question 2

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

Accepted Answer

A)    000 
B)    −22- \frac { \sqrt { 2 } } { 2 }−22c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
C)    22\frac { \sqrt { 2 } } { 2 }22c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
D)    2\sqrt { 2 }2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​ 
E)    −2- \sqrt { 2 }−2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​ 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)    3\sqrt { 3 }3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​   cos⁡θ\cos 	hetacosθ 
B)    cos⁡θ\cos 	hetacosθ 
C)    13cos⁡θ\frac { 1 } { 3 } \cos 	heta31​cosθ 
D)   0
E)   3  cos⁡θ\cos 	hetacosθ 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)    2\sqrt { 2 }2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​ 
B)    33\frac { \sqrt { 3 } } { 3 }33c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
C)    3\sqrt { 3 }3c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​ 
D)    22\frac { \sqrt { 2 } } { 2 }22c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
E)   1

Question 5

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

Accepted Answer

A)    1213\frac { 12 } { 13 }1312​ 
B)    512\frac { 5 } { 12 }125​ 
C)    513\frac { 5 } { 13 }135​ 
D)    1312\frac { 13 } { 12 }1213​ 
E)    135\frac { 13 } { 5 }513​

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)    12(1+sin⁡4∘) \frac { 1 } { 2 } \left( 1 + \sin 4 ^ { \circ } \right) 21​(1+sin4∘)  
B)    2cos⁡2∘\sqrt { 2 } \cos 2 ^ { \circ }2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​cos2∘ 
C)    12cos⁡4∘\frac { 1 } { 2 } \cos 4 ^ { \circ }21​cos4∘ 
D)    −12cos⁡4∘- \frac { 1 } { 2 } \cos 4 ^ { \circ }−21​cos4∘ 
E)    12(1−sin⁡4∘) \frac { 1 } { 2 } \left( 1 - \sin 4 ^ { \circ } \right) 21​(1−sin4∘)

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)    22\frac { \sqrt { 2 } } { 2 }22c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
B)   1/2
C)   1
D)   0
E)    32\frac { \sqrt { 3 } } { 2 }23c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​

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⁡(α+θ) =21221\cos ( \alpha + 	heta )  = \frac { 21 } { 221 }cos(α+θ) =22121​   cos⁡(α−θ) =−171221\cos ( \alpha - 	heta )  = - \frac { 171 } { 221 }cos(α−θ) =−221171​ 
B)    cos⁡(α+θ) =−5665cos⁡(α−θ) =1665\cos ( \alpha + 	heta )  = - \frac { 56 } { 65 } \cos ( \alpha - 	heta )  = \frac { 16 } { 65 }cos(α+θ) =−6556​cos(α−θ) =6516​ 
C)    cos⁡(α+θ) =1665\cos ( \alpha + 	heta )  = \frac { 16 } { 65 }cos(α+θ) =6516​   cos⁡(α−θ) =−5665\cos ( \alpha - 	heta )  = - \frac { 56 } { 65 }cos(α−θ) =−6556​ 
D)    cos⁡(α+θ) =5665\cos ( \alpha + 	heta )  = \frac { 56 } { 65 }cos(α+θ) =6556​   cos⁡(α−θ) =1665\cos ( \alpha - 	heta )  = \frac { 16 } { 65 }cos(α−θ) =6516​ 
E)    cos⁡(α+θ) =−171221\cos ( \alpha + 	heta )  = - \frac { 171 } { 221 }cos(α+θ) =−221171​   cos⁡(α−θ) =21221\cos ( \alpha - 	heta )  = \frac { 21 } { 221 }cos(α−θ) =22121​

Question 9

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

Accepted Answer

A)    120169\frac { 120 } { 169 }169120​ 
B)    −245389- \frac { 245 } { 389 }−389245​ 
C)    201389\frac { 201 } { 389 }389201​ 
D)    −119169- \frac { 119 } { 169 }−169119​ 
E)    −119120- \frac { 119 } { 120 }−120119​

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)    π2\frac { \pi } { 2 }2π​ 
C)   0
D)    π3\frac { \pi } { 3 }3π​ 
E)    2π3\frac { 2 \pi } { 3 }32π​

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)    32\frac { \sqrt { 3 } } { 2 }23c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
C)    −32- \frac { \sqrt { 3 } } { 2 }−23c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
D)   -1/2
E)    33\frac { \sqrt { 3 } } { 3 }33c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​

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)    −3105- \frac { 3 \sqrt { 10 } } { 5 }−5310c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
B)    3614\frac { 3 \sqrt { 6 } } { 14 }1436c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
C)    −8−374- \frac { \sqrt { 8 - 3 \sqrt { 7 } } } { 4 }−48−37c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
D)    −5+373- \frac { \sqrt { 5 + 3 \sqrt { 7 } } } { 3 }−35+37c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
E)    5−3103\frac { \sqrt { 5 - 3 \sqrt { 10 } } } { 3 }35−310c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​

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⁡(α−β) =140221\sin ( \alpha - \beta )  = \frac { 140 } { 221 }sin(α−β) =221140​   cos⁡(α−β) =−171221\cos ( \alpha - \beta )  = - \frac { 171 } { 221 }cos(α−β) =−221171​ 
B)    sin⁡(α−β) =−220221\sin ( \alpha - \beta )  = - \frac { 220 } { 221 }sin(α−β) =−221220​   cos⁡(α−β) =−21221\cos ( \alpha - \beta )  = - \frac { 21 } { 221 }cos(α−β) =−22121​ 
C)    sin⁡(α−β) =−171221\sin ( \alpha - \beta )  = - \frac { 171 } { 221 }sin(α−β) =−221171​   cos⁡(α−β) =140221\cos ( \alpha - \beta )  = \frac { 140 } { 221 }cos(α−β) =221140​ 
D)    sin⁡(α−β) =220221\sin ( \alpha - \beta )  = \frac { 220 } { 221 }sin(α−β) =221220​   cos⁡(α−β) =−21221\cos ( \alpha - \beta )  = - \frac { 21 } { 221 }cos(α−β) =−22121​ 
E)    sin⁡(α−β) =−21221\sin ( \alpha - \beta )  = - \frac { 21 } { 221 }sin(α−β) =−22121​   cos⁡(α−β) =−220221\cos ( \alpha - \beta )  = - \frac { 220 } { 221 }cos(α−β) =−221220​

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)    −1713- \frac { \sqrt { 17 } } { 13 }−1317c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
B)    1112\frac { \sqrt { 11 } } { 12 }1211c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
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)    θ≈75∘	heta \approx 75 ^ { \circ }θ≈75∘  , and  θ≈105∘	heta \approx 105 ^ { \circ }θ≈105∘ 
B)    θ≈195∘	heta \approx 195 ^ { \circ }θ≈195∘  , and  θ≈165∘	heta \approx 165 ^ { \circ }θ≈165∘ 
C)    θ≈75∘	heta \approx 75 ^ { \circ }θ≈75∘  , and  θ≈165∘	heta \approx 165 ^ { \circ }θ≈165∘ 
D)    θ≈15∘	heta \approx 15 ^ { \circ }θ≈15∘  , and  θ≈105∘	heta \approx 105 ^ { \circ }θ≈105∘ 
E)    θ≈150∘	heta \approx 150 ^ { \circ }θ≈150∘  , and  θ≈330∘	heta \approx 330 ^ { \circ }θ≈330∘  F)   θ≈195∘,andθ≈285∘	heta \approx 195 ^ { \circ } \quad, and  \quad 	heta \approx 285 ^ { \circ }θ≈195∘,andθ≈285∘ 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⁡(s+t2) 	an \left( \frac { s + t } { 2 } ight) tan(2s+t​)  
B)    tan⁡(s−t) 	an ( s - t ) tan(s−t)  
C)    sin⁡(s+t) \sin ( s + t ) sin(s+t)  
D)    cot⁡(s+t2) \cot \left( \frac { s + t } { 2 } ight) cot(2s+t​)  
E)    tan⁡(s−t2) 	an \left( \frac { s - t } { 2 } ight) tan(2s−t​)

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)    12(cos⁡3π5+1) \frac { 1 } { 2 } \left( \cos \frac { 3 \pi } { 5 } + 1 \right) 21​(cos53π​+1)  
B)    12(cos⁡3π5−1) \frac { 1 } { 2 } \left( \cos \frac { 3 \pi } { 5 } - 1 \right) 21​(cos53π​−1)  
C)    1+sin⁡2π51 + \sin ^ { 2 } \frac { \pi } { 5 }1+sin25π​ 
D)    −12sin⁡3π5- \frac { 1 } { 2 } \sin \frac { 3 \pi } { 5 }−21​sin53π​ 
E)    1−sin⁡2π51 - \sin ^ { 2 } \frac { \pi } { 5 }1−sin25π​

Question 18

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

Accepted Answer

A)   1/5
B)    51326\frac { 5 \sqrt { 13 } } { 26 }26513c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
C)    626\frac { \sqrt { 6 } } { 26 }266c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
D)   1/8
E)    2626\frac { \sqrt { 26 } } { 26 }2626c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​

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)    −1415- \frac { 14 } { 15 }−1514​ 
B)    −15414- \frac { \sqrt { 154 } } { 14 }−14154c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
C)    14314\frac { \sqrt { 143 } } { 14 }14143c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​ 
D)   4/15
E)    7715\frac { \sqrt { 77 } } { 15 }1577c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​​

Exam 9: Analytical Trigonometry

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

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

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 $\frac { 3 \pi } { 2 } < \theta < 2 \pi$

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∘

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

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. $\cos 47 ^ { \circ } \cos 43 ^ { \circ }$

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

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

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 $\frac { 3 \pi } { 2 } < \theta < 2 \pi$