Deck 34: Sum and Difference Formulas

Question 1

A weight is attached to a spring suspended vertically from a ceiling.When a driving force is applied to the system,the weight moves vertically from its equilibrium position,and this motion is modeled by​ $y = \frac { 1 } { 8 } \sin 2 t + \frac { 1 } { 6 } \cos 2 t$ ​ where y is the distance from equilibrium (in feet)and t is the time (in seconds).
​
Use the identity $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ where $C = \arctan ( b / a ) , a > 0$ ,to write the model in the form $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B t + C )$ .
​

A) $y = \sin ( 2 t + 0.9273 )$
B) $y = \frac { 24 } { 5 } \sin ( 2 t + 0.9273 )$ ​
C) $y = \frac { 24 } { 5 } \sin ( 2 t - 0.9273 )$
D) $y = \sin ( 2 t - 0.9273 )$
E) $y = \frac { 5 } { 24 } \sin ( 2 t + 0.9273 )$

Answer 1

$y = \frac { 5 } { 24 } \sin ( 2 t + 0.9273 )$

Question 2

Find the expression as the sine of an angle.​ $$\sin 55 ^ { \circ } \cos 5 ^ { \circ } + \cos 55 ^ { \circ } \sin 5 ^ { \circ }$$ ​

A) $\sin 55 ^ { \circ }$
B) $\cos 60 ^ { \circ }$
C) $\cos 50 ^ { \circ }$
D) $\sin 60 ^ { \circ }$
E) $\sin 50 ^ { \circ }$

Answer 2

$\sin 60 ^ { \circ }$

Question 3

Find the expression as the sine or cosine of an angle.​ $$\cos 100 ^ { \circ } \cos 60 ^ { \circ } - \sin 100 ^ { \circ } \sin 60 ^ { \circ }$$ ​

A) $\cos 40 ^ { \circ }$
B) $\sin 40 ^ { \circ }$
C) $\sin 160 ^ { \circ }$
D) $\cos 160 ^ { \circ }$
E) $\cos 100 ^ { \circ }$

Answer 3

$\cos 160 ^ { \circ }$

Question 4

Simplify the expression algebraically.​ $$3 \cos ( \pi - \theta ) + 3 \sin \left( \frac { \pi } { 2 } + \theta \right)$$ ​

A) $3 \cos ( \theta ) - 3 \sin ( \theta )$
B)0
C) $3 \cos ( \theta ) + 3 \sin ( \theta )$
D)1
E)6

Question 5

Simplify the expression algebraically.​ $$\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$$ ​

A) $\cos ^ { 2 } 7 x - \sin ^ { 2 } y$
B) $\cos ^ { 2 } 7 x + \sin ^ { 2 } 4 y$
C) $\cos ^ { 2 } 7 x - \sin ^ { 2 } 4 y$
D) $\cos ^ { 2 } x - \sin ^ { 2 } 4 y$
E) $\cos ^ { 2 } x - \sin ^ { 2 } y$

Question 6

Find the expression as the tangent of an angle.​ $$\frac { \tan 3 x + \tan x } { 1 - \tan 3 x \tan x }$$ ​

A) $\tan 2 x$
B) $\tan 3 x$
C) $\tan 4 x$
D) $\tan ^ { - 1 } 4 x$
E) $\tan ^ { - 1 } 2 x$

Question 7

Simplify the expression algebraically. $$4 \tan \left( \frac { \pi } { 4 } - \theta \right)$$ ​

A) $\frac { 4 - 4 \tan \theta } { 1 + \tan \theta }$
B) $\frac { 4 - 4 \tan \theta } { \tan \theta }$
C) $\frac { \tan \theta } { 4 - \tan \theta }$
D) $\frac { 4 + 4 \tan \theta } { 1 - \tan \theta }$
E) $\frac { 4 + 4 \tan \theta } { \tan \theta }$

Question 8

Simplify the expression algebraically.​ $$\sin ( 9 x + 9 y ) \sin ( 9 x - 9 y )$$ ​

A) $\sin ^ { 2 } x - \sin ^ { 2 } 9 y$
B) $\sin ^ { 2 } 9 x + \sin ^ { 2 } 9 y$
C) $\sin ^ { 2 } 9 x - \sin ^ { 2 } 9 y$
D) $\sin ^ { 2 } x + \sin ^ { 2 } y$
E) $\sin ^ { 2 } 9 x - \sin ^ { 2 } y$

Question 9

Simplify the expression algebraically.​ $$\frac { 9 } { \sqrt { 2 } } \cos \left( \frac { 5 \pi } { 4 } - x \right)$$ ​

A)- $\frac { 9 } { 2 }$ $( \cos x + \sin x )$
B) $\frac { 9 } { 2 }$ $( \cos x - \sin x )$
C) $\frac { 9 } { 2 }$ $\left( \cos \frac { 5 x } { 4 } + \sin \frac { 5 x } { 4 } \right)$
D) $\frac { 9 } { 2 }$ $( \sin x - \cos x )$
E) $\frac { 9 } { 2 }$ $\left( \cos \frac { 5 x } { 4 } - \sin \frac { 5 x } { 4 } \right)$

Question 10

Find the expression as the sine or cosine of an angle.​ $$\cos 9 x \cos 5 y + \sin 9 x \sin 5 y$$ ​

A) $\sin ( 5 x - 9 y )$
B) $\sin ( 9 x - 5 y )$
C) $\cos ( 5 x - 9 y )$
D) $\cos ( 9 x - 5 y )$
E) $\cos ( 9 x + 5 y )$

Question 11

Find the expression as the tangent of an angle.​ $$\frac { \tan 130 ^ { \circ } - \tan 30 ^ { \circ } } { 1 + \tan 130 ^ { \circ } \tan 30 ^ { \circ } }$$ ​

A) $\tan ^ { - 1 } 100 ^ { \circ }$
B) $\tan ^ { - 1 } 130 ^ { \circ }$
C) $\tan 160 ^ { \circ }$
D) $\tan 100 ^ { \circ }$
E) $\tan 30 ^ { \circ }$

Question 12

Simplify the expression algebraically.​ $\cos ( 6 x + 9 y ) + \cos ( 6 x - 9 y )$ ​

A) $\cos 6 x$
B) $\cos 6 x \cos 9 y$
C) $2 \cos 6 x \cos 9 y$
D) $2 \cos 6 x$
E) $2 \cos x \cos y$

Question 13

Simplify the expression algebraically. $$5 \sin \left( \frac { \pi } { 6 } + x \right)$$ ​

A) $\frac { 5 } { 2 }$ $( \cos x - \sqrt { 3 } \sin x )$
B) $\frac { 5 } { 2 }$ $( \cos x + \sqrt { 3 } \sin x )$
C) $\frac { 5 } { 2 }$ $( \sin x - \sqrt { 3 } \cos x )$
D) $\frac { 5 } { 2 }$ $( \cos x + \sin x )$
E) $\frac { 5 } { 2 }$ $( \sin x + \sqrt { 3 } \cos x )$

Question 14

A weight is attached to a spring suspended vertically from a ceiling.When a driving force is applied to the system,the weight moves vertically from its equilibrium position,and this motion is modeled by​ $$y = \frac { 1 } { 8 } \sin 2 t + \frac { 1 } { 6 } \cos 2 t$$ where y is the distance from equilibrium (in feet)and t is the time (in seconds). Find the amplitude of the oscillations of the weight.
​

A) $\frac { 1 } { 24 } \mathrm { ft }$
B) $\frac { 1 } { 10 } \mathrm { ft }$
C) $\frac { 5 } { 24 } \mathrm { ft }$
D) $\frac { 1 } { 5 } \mathrm { ft }$
E) $\frac { 24 } { 5 } \mathrm { ft }$

Question 15

Find the expression as the tangent of an angle.​ $$\frac { \tan 60 ^ { \circ } - \tan 20 ^ { \circ } } { 1 + \tan 60 ^ { \circ } \tan 20 ^ { \circ } }$$ ​

A) $\tan 40 ^ { \circ }$
B) $\tan 60 ^ { \circ }$
C) $\tan ^ { - 1 } 80 ^ { \circ }$
D) $\tan 20 ^ { \circ }$
E) $\tan ^ { - 1 } 40 ^ { \circ }$

Question 16

Simplify the expression algebraically.​ $$6 \sin \left( \frac { \pi } { 2 } + x \right)$$ ​

A) $6 \cos x$
B) $- 6 \cos x$
C) $6 \sin x$
D) $- \frac { 1 } { 6 } \cos x$
E) $\frac { 1 } { 6 } \cos x$

Question 17

Simplify the expression algebraically.​ $$3 \sin \left( \frac { \pi } { 2 } - x \right)$$ ​

A) $\frac { 1 } { 3 } \cos x$
B) $3 \cos x$
C) $- \frac { 1 } { 3 } \cos x$
D) $- 3 \cos x$
E) $3 \sin x$

Question 18

Find the expression as the cosine of an angle. ​​ $$\cos \frac { \pi } { 5 } \cos \frac { \pi } { 3 } - \sin \frac { \pi } { 5 } \sin \frac { \pi } { 3 }$$ ​

A) $\sin \frac { 15 \pi } { 8 }$
B) $\cos \frac { 15 \pi } { 8 }$
C) $\cos \frac { 8 \pi } { 15 }$
D) $\cos \frac { \pi } { 15 }$
E) $\sin \frac { 8 \pi } { 15 }$

Question 19

Simplify the expression algebraically.​ $$\sin ( 7 x + 7 y ) + \sin ( 7 x - 7 y )$$ ​

A) $\sin \left( 7 x ^ { 2 } + 7 y ^ { 2 } \right)$
B) $2 \sin 7 x$
C) $\sin \left( 7 x ^ { 2 } - 7 y ^ { 2 } \right)$
D) $\sin 7 x \cos 7 y$
E) $2 \sin 7 x \cos 7 y$

Question 20

Find the expression as the sine of an angle.​ $$\sin 5 \cos 1.7 - \cos 5 \sin 1.7$$ ​

A) $\sin 3.4$
B) $\sin 3.3$
C) $\sin 3.5$
D) $\sin 3.7$
E) $\sin 3.6$

Question 21

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ,where $C = \arctan ( a / b ) , a = 9 , b = 2 , B = 2$ to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A) $\sqrt { 85 }$ $\cos ( 3 \theta - 1.3521 )$
B)9 $\cos ( 3 \theta + 1.3521 )$
C)9 $\cos ( 3 \theta - 1.3521 )$
D) $\sqrt { 85 }$ $\cos ( 3 \theta + 1.3521 )$
E)2 $\cos ( 3 \theta - 1.3521 )$

Question 22

Simplify the following expression algebraically.​ $$6 \tan ( \pi + \theta )$$ ​

A) $$- \frac { 3 } { 2 } \tan \theta$$
B) $$- 6 \sin \theta$$
C) $- 6 \tan \theta$
D) $6 \tan \theta$
E) $\frac { 3 } { 2 } \tan \theta$

Question 23

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ,where $C = \arctan ( b / a ) , a = 5 , b = 8 , B = 1$ ,to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A) $\sqrt { 89 }$ $\sin ( \theta + 1.0122 )$
B) $\sqrt { 89 }$ $\sin ( \theta - 1.0122 )$
C) $\sqrt { 89 }$ $\sin ( 2 \theta - 1.0122 )$
D) $\sqrt { 89 }$ $\sin ( 2 \theta + 1.0122 )$
E) $\sin ( 2 \theta + 1.0122 )$

Question 24

Simplify the following expression algebraically.​ $$4 \sin \left( \frac { 3 \pi } { 2 } + \theta \right)$$ ​

A) $\frac { 3 } { 2 } \cos \theta$
B) $4 \cos \theta$
C) $- 4 \sin \theta$
D) $- \frac { 3 } { 2 } \cos \theta$
E) $- 4 \cos \theta$

Question 25

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ,where $C = \arctan ( b / a ) , a = 18 , b = 6 , B = 3$ ,to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A) $6 \sqrt { 10 }$ $\sin ( \theta - 0.3218 )$
B) $6 \sqrt { 10 }$ $\sin ( 3 \theta + 0.3218 )$
C) $\sin ( 3 \theta + 0.3218 )$
D) $6 \sqrt { 10 }$ $\sin ( 3 \theta - 0.3218 )$
E) $6 \sqrt { 10 }$ $\sin ( \theta + 0.3218 )$

Question 26

Simplify the following expression algebraically.​ $7 \cos ( \pi + x )$ ​

A) $- 7 \sin x$
B) $- 7 \cos x$
C) $7 \cos x$
D) $\frac { 3 } { 2 } \cos x$
E) $- \frac { 3 } { 2 } \cos x$

Question 27

Use a graphing utility to select correct graph of $y _ { 1 }$ and $y _ { 2 }$ in the same viewing window.Use the graphs to determine whether $y _ { 1 } = y _ { 2 }$ .Explain your reasoning.​ $$y _ { 1 } = \sin ( x + 6 ) , y _ { 2 } = \sin ( x ) + \sin ( 6 )$$ ​

A)​ No, $$y _ { 1 } = y _ { 2 }$$ because their graphs are different.
B)​ Yes, $$y _ { 1 } = y _ { 2 }$$ because their graphs are different.
C)​ Yes, $$y _ { 1 } = y _ { 2 }$$ because their graphs are same.
D)​ No, $$y _ { 1 } \neq y _ { 2 }$$ because their graphs are Same.
E)​ No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are different.

Question 28

Use the formula $$a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$$ ,where $C = \arctan ( b / a ) , a = 3 , b = , B = 1$ to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A) $\sin ( \theta + 0.3218 )$
B) $\sin ( \theta - 0.3218 )$
C)3 $\sin ( \theta + 0.3218 )$
D) $\sqrt { 10 }$ $\sin ( \theta - 0.3218 )$
E) $\sqrt { 10 }$ $\sin ( \theta + 0.3218 )$

Question 29

Simplify the following expression algebraically.​ $$\cos \left( \frac { 3 \pi } { 2 } - x \right)$$ ​

A) $\sin x$
B) $\frac { 3 } { 2 } \sin x$
C) $- \cos x$
D) $- \sin x$
E) $- \frac { 3 } { 2 } \sin x$

Question 30

Use a graphing utility to select the correct graph of ​ $y _ { 1 }$ and $y _ { 2 }$ in the same viewing window.Use the graphs to determine whether $y _ { 1 } = y _ { 2 }$ .Explain your reasoning.​ $y _ { 1 } = \cos ( x + 4 ) , y _ { 2 } = \cos x + \cos 4$ ​

A)​ No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are different.
B)​ No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are Same.
C)​ Yes, $y _ { 1 } = y _ { 2 }$ because their graphs are different.
D)​ Yes, $y _ { 1 } = y _ { 2 }$ because their graphs are same.
E)​ No, $y _ { 1 } = y _ { 2 }$ because their graphs are different.

Question 31

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ,where $C = \arctan ( a / b ) , a = 13 , b = 6 , B = 3$ to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A)6 $\cos ( 3 \theta - 1.1384 )$
B) $\sqrt { 205 }$ $\cos ( 3 \theta + 1.1384 )$
C) $\sqrt { 205 }$ $\cos ( 3 \theta - 1.1384 )$
D)13 $\cos ( 3 \theta + 1.1384 )$
E)13 $\cos ( 3 \theta - 1.1384 )$

Question 32

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ,where $C = \arctan ( a / b ) , a > 0$ ,to rewrite the trigonometric expression in the form.​ $\sqrt { 6 } \sin \left( \theta + \frac { \pi } { 4 } \right)$ ​

A) $$\sqrt { 3 } \sin \theta - \sqrt { 3 } \cos \theta$$
B) $$\sqrt { 3 } \sin \theta + \cos \theta$$
C) $\sin \theta - \sqrt { 3 } \cos \theta$
D) $\sqrt { 3 } \sin \theta + \sqrt { 3 } \cos \theta$
E) $\sin \theta + \cos \theta$

Question 33

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ,where $C = \arctan ( a / b )$ $C = \arctan ( a / b ) , a > 0$ ,to rewrite the trigonometric expression in the form $a \sin B \theta + b \cos B \theta$ ​ 9 $\cos \left( \theta - \frac { \pi } { 4 } \right)$ ​

A) $\frac { 9 \sqrt { 2 } } { 2 } \cos \theta$
B) $$- \frac { 9 \sqrt { 2 } } { 2 } \sin \theta - \frac { 9 \sqrt { 2 } } { 2 } \cos \theta$$
C) $$- \frac { 9 \sqrt { 2 } } { 2 } \sin \theta + \frac { 9 \sqrt { 2 } } { 2 } \cos \theta$$
D) $\frac { 9 \sqrt { 2 } } { 2 } \sin \theta + \frac { 9 \sqrt { 2 } } { 2 } \cos \theta$
E) $\frac { 9 \sqrt { 2 } } { 2 } \sin \theta - \frac { 9 \sqrt { 2 } } { 2 } \cos \theta$

Question 34

Find the exact value of the given expression using a sum or difference formula. $$\sin 285 ^ { \circ }$$

A) $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
B) $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$

Question 35

Find the exact value of the given expression. $$\sin \left( \frac { \pi } { 3 } - \frac { \pi } { 4 } \right)$$

A) $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
B) $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$

Question 36

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta - C )$ ,where $C = \arctan ( a / b ) , a = 2 , b = 8 , B = 1$ ,to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta - C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A)2 $\cos ( \theta + 0.245 )$
B) $2 \sqrt { 17 }$ $\cos ( \theta - 0.245 )$
C) $2 \sqrt { 17 }$ $\cos ( \theta + 0.245 )$
D)2 $\cos ( \theta - 0.245 )$
E)8 $\cos ( \theta - 0.245 )$

Question 37

Find the exact value of the given expression. $$\cos \left( 120 ^ { \circ } + 315 ^ { \circ } \right)$$

A) $\frac { ( 1 - \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$
B) $\frac { ( - 1 + \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( 1 + \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { ( - 1 - \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$

Question 38

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ,where $C = \arctan ( a / b ) , a = 3 , b = 7 , B = 2$ to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A) $\sqrt { 58 }$ $\cos ( 2 \theta + 0.4049 )$
B) $\sqrt { 58 }$ $\cos ( 2 \theta - 0.4049 )$
C)7 $\cos ( 2 \theta - 0.4049 )$
D)3 $\cos ( 2 \theta - 0.4049 )$
E)3 $\cos ( 2 \theta + 0.4049 )$

Question 39

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ,where $C = \arctan ( b / a ) , a = 1 , b = 3 , B = 2$ ,to rewrite the trigonometric expression in the following form.​ $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ ​ $a \sin B \theta + b \cos B \theta$ ​

A) $\sqrt { 10 }$ $\sin ( 2 \theta + 1.249 )$
B) $\sqrt { 10 }$ $\sin ( \theta - 1.249 )$
C) $\sqrt { 10 }$ $\sin ( 2 \theta - 1.249 )$
D) $\sqrt { 10 }$ $\sin ( \theta + 1.249 )$
E) $\sin ( 2 \theta + 1.249 )$

Question 40

Find the exact value of the given expression using a sum or difference formula. $$\cos \frac { 17 \pi } { 12 }$$

A) $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
B) $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$

Question 41

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 3 } { 5 }$ and $\cos v = - \frac { 24 } { 25 }$ .(Both u and v are in Quadrant II. )

A) $\sin ( u + v ) = - \frac { 44 } { 125 }$
B) $\sin ( u + v ) = - \frac { 3 } { 5 }$
C) $\sin ( u + v ) = \frac { 4 } { 5 }$
D) $\sin ( u + v ) = - \frac { 4 } { 5 }$
E) $\sin ( u + v ) = \frac { 44 } { 125 }$

Question 42

Find the exact value of the given expression.​ $\cos \left( 300 ^ { \circ } + 135 ^ { \circ } \right)$

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

Question 43

Simplify the given expression algebraically. $\cos \left( \frac { \pi } { 2 } + x \right)$

A) $\cos x$
B) $- \sin x$
C) $\sin x$
D)1
E) $- \cos x$

Question 44

Write the given expression as the cosine of an angle.​ $$\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$$

A) $\cos \left( 45 ^ { \circ } \right)$
B) $\cos \left( 5 ^ { \circ } \right)$
C) $\cos \left( - 80 ^ { \circ } \right)$
D) $\cos \left( 40 ^ { \circ } \right)$
E) $\cos \left( 85 ^ { \circ } \right)$

Question 45

Find the exact value of the given expression. $\sin \left( \frac { 2 \pi } { 3 } - \frac { 5 \pi } { 4 } \right)$

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

Question 46

Write the given expression as the tangent of an angle. $\frac { \tan 3 x + \tan 4 x } { 1 - \tan 3 x \tan 4 x }$

A) $\tan ( 7 x )$
B) $\tan ( 3 x )$
C) $\tan ( 5 x )$
D) $\tan ( - x )$
E) $\tan ( - 5 x )$

Question 47

Find the exact solutions of the given equation in the interval [0,2π). sin 2x = sin x
​

A) $x = \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$
B) $x = 0 , \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$
C) $x = \frac { \pi } { 2 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$
D) $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$
E) $x = 0$

Question 48

Simplify the given expression algebraically.​ $\cos ( x - \pi )$

A) $\sin x$
B) $\cos x$
C) $- \sin x$
D) $- \cos x$
E)1

Question 49

Write the given expression as the cosine of an angle. $\cos 60 ^ { \circ } \cos 25 ^ { \circ } - \sin 60 ^ { \circ } \sin 25 ^ { \circ }$ ​

A) $\cos \left( 25 ^ { \circ } \right)$
B) $\cos \left( 85 ^ { \circ } \right)$
C) $\cos \left( 35 ^ { \circ } \right)$
D) $\cos \left( 60 ^ { \circ } \right)$
E) $\cos \left( - 50 ^ { \circ } \right)$

Question 50

Write the given expression as the sine of an angle. $$\sin 85 ^ { \circ } \cos 50 ^ { \circ } - \sin 50 ^ { \circ } \cos 85 ^ { \circ }$$

A) $\sin \left( 135 ^ { \circ } \right)$
B) $\sin \left( - 100 ^ { \circ } \right)$
C) $\sin \left( 35 ^ { \circ } \right)$
D) $\sin \left( 50 ^ { \circ } \right)$
E) $\sin \left( 85 ^ { \circ } \right)$

Question 51

Write the given expression as an algebraic expression. cos(arccos x - arcsin x)

A) $0$
B) $2 x \sqrt { 1 - x ^ { 2 } }$
C) $\frac { \left( x \sqrt { 1 - x ^ { 2 } } - x \right) \left( \sqrt { x ^ { 2 } + 1 } \right) } { x ^ { 2 } + 1 }$
D)1
E) $\frac { \left( x \sqrt { 1 - x ^ { 2 } } + x \right) \left( \sqrt { x ^ { 2 } + 1 } \right) } { x ^ { 2 } + 1 }$

Question 52

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 8 } { 17 }$ and $\cos v = - \frac { 60 } { 61 }$ .(Both u and v are in Quadrant II. )

A) $\cos ( u + v ) = \frac { 812 } { 1037 }$
B) $\cos ( u + v ) = - \frac { 300 } { 1037 }$
C) $\cos ( u + v ) = - \frac { 804 } { 1037 }$
D) $\cos ( u + v ) = \frac { 315 } { 1037 }$
E) $\cos ( u + v ) = \frac { 645 } { 1037 }$

Question 53

Find the exact value of the given expression. $$\sin 105 ^ { \circ } \cos 345 ^ { \circ } - \sin 345 ^ { \circ } \cos 105 ^ { \circ }$$

A) $\frac { 1 } { 2 }$
B) $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) $\frac { \sqrt { 3 } } { 2 }$

Question 54

Find the exact value of $\sin ( u + v )$ given that $\sin u = \frac { 7 } { 25 }$ and $\cos v = - \frac { 12 } { 13 }$ .(Both u and v are in Quadrant II. )

A) $\sin ( u + v ) = \frac { 36 } { 325 }$
B) $\sin ( u + v ) = - \frac { 36 } { 325 }$
C) $\sin ( u + v ) = - \frac { 204 } { 325 }$
D) $\sin ( u + v ) = - \frac { 253 } { 325 }$
E) $\sin ( u + v ) = \frac { 204 } { 325 }$

Question 55

Write the given expression as the tangent of an angle.​ $$\frac { \tan 7 x + \tan 4 x } { 1 - \tan 7 x \tan 4 x }$$

A) $\tan ( - 17 x )$
B) $\tan ( 11 x )$
C) $\tan ( 7 x )$
D) $\tan ( 3 x )$
E) $\tan ( 5 x )$

Question 56

Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$

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

Question 57

Write the given expression as the sine of an angle. $\sin 45 ^ { \circ } \cos 55 ^ { \circ } - \sin 55 ^ { \circ } 45 ^ { \circ }$

A) $\sin \left( - 110 ^ { \circ } \right)$
B) $\sin \left( - 10 ^ { \circ } \right)$
C) $\sin \left( 100 ^ { \circ } \right)$
D) $\sin \left( 45 ^ { \circ } \right)$
E) $\sin \left( 55 ^ { \circ } \right)$

Question 58

Evaluate the expression.​ $\sin x \sin ( x + y ) + \cos x \cos ( x + y )$ ​

A) $\sin ^ { 2 } x - \sin ^ { 2 } y$
B)0
C) $\sin x \cos x + \sin y \cos y$
D) $\cos y$
E) $\cos ^ { 2 } x - \sin ^ { 2 } x$

Question 59

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 5 } { 13 }$ and $\cos v = - \frac { 4 } { 5 }$ .(Both u and v are in Quadrant II. )

A) $\cos ( u + v ) = \frac { 28 } { 65 }$
B) $\cos ( u + v ) = - \frac { 16 } { 65 }$
C) $\cos ( u + v ) = \frac { 33 } { 65 }$
D) $\cos ( u + v ) = - \frac { 28 } { 65 }$
E) $\cos ( u + v ) = \frac { 56 } { 65 }$

Question 60

Find the exact value of the given expression using a sum or difference formula. $\cos \frac { 13 \pi } { 12 }$

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

Question 61

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ . sin 4x= -2sin 2x

A) $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$
B) $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$
C) $x = \frac { \pi } { 6 } , \frac { \pi } { 2 } , \frac { 5 \pi } { 6 } , \frac { 3 \pi } { 2 }$
D) $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$
E) $x = \frac { 7 \pi } { 6 } , \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 }$

Question 62

Verify the given identity.