Deck 5: Analytic Trigonometry

Question 1

Use the figure to find the exact value of the trigonometric function.
Sec 2 $\theta$

a = 1, b = 8

A) $\frac { 63 } { 64 }$
B) $\frac { 63 } { 65 }$
C) $\frac { 64 } { 65 }$
D) $\frac { 65 } { 63 }$
E) $\frac { 65 } { 64 }$

Answer 1

$\frac { 65 } { 63 }$

Question 2

Use the half-angle formulas to simplify the expression.
$\sqrt { \frac { 1 - \cos 10 x } { 2 } }$

A)|sin 5x|
B)- |sin x|
C)|sin 10x|
D)- |sin 5x|
E)|sin x|

Answer 2

|sin 5x|

Question 3

Use the figure to find the exact value of the trigonometric function.
Sin 2 $\theta$

A = 1, b = 2

A) $\frac { 4 } { 5 }$
B) $\frac { 4 } { 5 }$ ..
C) $\frac { 5 } { 4 }$
D) $\frac { 5 } { 5 }$
E) $\frac { 5 } { 5 }$ ..

Answer 3

$\frac { 4 } { 5 }$ ..

Question 4

Use the figure to find the exact value of the trigonometric function.
Cot 2 $\theta$

a = 1, b = 6

A) $\frac { 12 } { 35 }$
B) $\frac { 35 } { 37 }$
C) $\frac { 12 } { 37 }$
D) $\frac { 37 } { 35 }$
E) $\frac { 35 } { 12 }$

Question 5

Use a double-angle formula to rewrite the expression. ​
3 - 6 sin² x
​

A)6 cos x
B)​3 sin 2x
C)​3 sin x
D)​3 cos 2x
E)​6 cos 2x

Question 6

Use the half-angle formulas to simplify the expression. $- \sqrt { \frac { 1 + \cos 6 x } { 1 - \cos 6 x } }$

A) - |cot x|
B) - |3 tan x|
C) - |3 tan 6x|
D) - |3 cot 3x|
E) - |cot 3x|

Question 7

Use the figure to find the exact value of the trigonometric function.
Tan 2 $\theta$

a = 1, b = 6

A) $\frac { 35 } { 12 }$
B) $\frac { 12 } { 35 }$
C) $\frac { 12 } { 37 }$
D) $\frac { 35 } { 37 }$
E) $\frac { 37 } { 35 }$

Question 8

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\sin 3 \theta - \sin \theta$

A) $2 \sin 3 \theta \cos \theta$
B) $2 \sin 2 \theta \cos \theta$
C) $2 \cos 2 \theta \cos \theta$
D) $2 \sin 2 \theta \sin \theta$
E) $2 \cos 2 \theta \sin \theta$

Question 9

Use a double-angle formula to rewrite the expression. ​
2 sin² x - 1
​

A)cos x
B)cos 2x​
C)-2 cos x​
D)2 cos 2x​
E)- cos 2x​

Question 10

Use the figure to find the exact value of the trigonometric function.
Csc 2 $\theta$

a = 1, b = 6

A) $\frac { 37 } { 12 }$
B) $\frac { 13 } { 37 }$
C) $\frac { 37 } { 13 }$
D) $\frac { 12 } { 13 }$
E) $\frac { 12 } { 37 }$

Question 11

Use the half-angle formulas to simplify the expression. $- \sqrt { \frac { 1 - \cos ( x - 3 ) } { 2 } }$

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

Question 12

Use the product-to-sum formulas to rewrite the product as a sum or difference. $10 \cos 45 ^ { \circ } \cos 20 ^ { \circ }$

A) $10 \left( \cos 25 ^ { \circ } + \cos 65 ^ { \circ } \right)$
B) $\cos 25 ^ { \circ } + \cos 65 ^ { \circ }$
C) $5 \left( \cos 65 ^ { \circ } - \cos 25 ^ { \circ } \right)$
D) $5 \left( \cos 25 ^ { \circ } + \cos 65 ^ { \circ } \right)$
E) $5 \left( \cos 65 ^ { \circ } + \sin 25 ^ { \circ } \right)$

Question 13

Use the product-to-sum formulas to rewrite the product as a sum or difference. $\sin \frac { \pi } { 3 } \cos \frac { \pi } { 6 }$

A) $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 6 } \right)$
B) $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } + \sin \frac { \pi } { 6 } \right)$
C) $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } - \cos \frac { \pi } { 6 } \right)$
D) $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } - \sin \frac { \pi } { 6 } \right)$
E) $\left( \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 6 } \right)$

Question 14

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\sin 9 \theta + \sin 7 \theta$

A) $2 \sin 8 \theta \sin \theta$
B) $2 \cos 8 \theta \cos \theta$
C) $2 \sin 8 \theta \cos \theta$
D) $2 \cos 8 \theta \sin \theta$
E) $2 \sin 9 \theta \cos 7 \theta$

Question 15

Use a double-angle formula to rewrite the expression. ​
10 cos² x - 5
​

A)5 cos x
B)​cos 5x
C)​10 cos 2x
D)​10 cos x
E)​5 cos 2x

Question 16

Use the product-to-sum formulas to rewrite the product as a sum or difference. $8 \sin 65 ^ { \circ } \cos 25 ^ { \circ }$

A) $8 \left( \sin 90 ^ { \circ } + \sin 40 ^ { \circ } \right)$
B) $4 \left( \cos 90 ^ { \circ } + \cos 40 ^ { \circ } \right)$
C) $4 \left( \cos 90 ^ { \circ } - \cos 40 ^ { \circ } \right)$
D) $\left( \sin 90 ^ { \circ } + \sin 40 ^ { \circ } \right)$
E) $4 \left( \sin 90 ^ { \circ } + \sin 40 ^ { \circ } \right)$

Question 17

Use the figure to find the exact value of the trigonometric function.
Cos 2 $\theta$

a = 1, b = 2

A) $\frac { 3 } { 4 }$
B) $\frac { 3 } { 5 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 4 } { 5 }$
E) $\frac { 5 } { 4 }$

Question 18

Use a double-angle formula to rewrite the expression.
$2 \sin x \cos x$

A) sin x
B) 2 sin x
C) 2 sin 2x.
D) sin x..
E) sin 2x

Question 19

Use the half-angle formulas to simplify the expression.
$\sqrt { \frac { 1 + \cos 8 x } { 2 } }$

A) - |cos x|
B) |cos x|
C) |cos 4x|
D)- |cos 4x|
E) - |cos 8x|

Question 20

Use the product-to-sum formulas to rewrite the product as a sum or difference. $4 \cos \frac { \pi } { 2 } \sin \frac { 5 \pi } { 4 }$

A) $2 \left( \sin \frac { 7 \pi } { 4 } - \sin \frac { 3 \pi } { 4 } \right)$
B) $2 \left( \cos \frac { 7 \pi } { 4 } - \cos \frac { 3 \pi } { 4 } \right)$
C) $2 \left( \sin \frac { 7 \pi } { 4 } + \cos \frac { 3 \pi } { 4 } \right)$
D) $2 \left( \sin \frac { 7 \pi } { 4 } + \sin \frac { 3 \pi } { 4 } \right)$
E) $2 \left( \cos \frac { 7 \pi } { 4 } + \cos \frac { 3 \pi } { 4 } \right)$

Question 21

Find the exact solutions of the given equation in the interval [0, 2 $\pi$ ).
$2 \sin ^ { 2 } x + 3 \sin x = - 1$

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

Question 22

Use the figure below to determine the exact value of the given function. $\csc 2 \theta$

A) $\csc 2 \theta = \frac { 13 } { 5 }$
B) $\csc 2 \theta = \frac { 13 } { 7 }$
C) $\csc 2 \theta = \frac { 13 } { 5 }$ .
D) $\csc 2 \theta = \frac { 13 } { 9 }$
E) $\csc 2 \theta = \frac { 13 } { 12 }$

Question 23

Convert the expression.
$( 2 \sin x + 2 \cos x ) ^ { 2 }$

A) $4 - 4 \sin 2 x$
B) $2 + 4 \sin x$
C) $4 + 4 \sin x$
D) $4 - 4 \sin x$
E) $4 + 4 \sin 2 x$

Question 24

Convert the expression. $\tan \frac { a } { 2 }$

A) $\csc a - \tan a$
B) $\csc a + \cot a$
C) $\cos a - \cot a$
D) $\cos a + \cot a$
E) $\csc a - \cot a$

Question 25

Convert the expression. $\cos 4 \alpha$

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

Question 26

Convert the expression. $3 \csc 2 \theta$

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

Question 27

Use a double-angle formula to find the exact value of cos2u when $$\sin u = \frac { 7 } { 25 } \text {, where } \frac { \pi } { 2 } < u < \pi$$

A) $\cos 2 u = \frac { 527 } { 625 }$
B) $\cos 2 u = - \frac { 1152 } { 625 }$
C) $\cos 2 u = \frac { 336 } { 625 }$
D) $\cos 2 u = \frac { 168 } { 625 }$
E) $\cos 2 u = - \frac { 478 } { 625 }$

Question 28

Convert the expression.
$1 + \cos 4 y$

A) $2 \cos ^ { 2 } 2 y$
B) $\cos ^ { 2 } 4 y$
C) $\cos ^ { 2 } 2 y$
D) $4 \cos ^ { 2 } 4 y$
E) $2 \cos ^ { 2 } 4 y$

Question 29

Use the figure to find the exact value of the trigonometric function. $$\begin{array} { l } a = 12 , b = 3 \\c = 5 , d = 4\end{array}$$
Cos 2 $\beta$

A) $\frac { 41 } { 5 }$
B) $\frac { 3 } { 10 }$
C) $\frac { 9 } { 41 }$
D) $\frac { 12 } { 13 }$
E) $\frac { 41 } { 9 }$

Question 30

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\cos 12 \theta + \cos 8 \theta$

A) $2 \sin 10 \theta \cos 2 \theta$
B) $2 \cos 10 \theta \sin 2 \theta$
C) $2 \sin 12 \theta \cos 8 \theta$
D) $2 \cos 10 \theta \cos 2 \theta$
E) $2 \sin 10 \theta \sin 2 \theta$

Question 31

Use a double angle formula to rewrite the given expression. $10 \cos ^ { 2 } x - 5$

A) $5 \cos 5 x$
B) $5 \cos 2 x$
C) $10 \cos 2 x$
D) $2 \cos 10 x$
E) $2 \cos 5 x$

Question 32

Use the sum-to-product formulas to rewrite the sum or difference as a product.
Cos 3 $\theta$ + cos 8 $\theta$

A) $\cos \frac { 11 \theta } { 2 } \cos \frac { 5 \theta } { 2 }$
B) $2 \cos \frac { 11 \theta } { 2 } \sin - \frac { 5 \theta } { 2 }$
C) $2 \sin \frac { 11 \theta } { 2 } \sin \frac { 5 \theta } { 2 }$
D) $\cos - \frac { 11 \theta } { 2 } \cos \frac { 5 \theta } { 2 }$
E) $2 \cos \frac { 11 \theta } { 2 } \cos - \frac { 5 \theta } { 2 }$

Question 33

Convert the expression. $\cos ^ { 4 } b - \sin ^ { 4 } b$

A)2 cos b
B) cos 2b
C) cos b
D)2 cos 2b
E) 4 cos b

Question 34

When two railroad tracks merge, the overlapping portions of the tracks are in the shapes of circular arcs (see figure).The radius of each arc r (in feet) and the angle $\theta$ are related by
$$\frac { x } { 2 } = 2 r \sin ^ { 2 } \frac { \theta } { 2 }$$
Write a formula for x in terms of cos $\theta$ .

A) $x = r \left( 1 - \cos \frac { \theta } { 2 } \right)$
B) $x = r ( 1 - \cos \theta )$
C) $x = 2 r ( 1 - \cos \theta )$
D) $x = 2 r ( 1 + \cos \theta )$
E) $x = 2 r \left( 1 - \cos \frac { \theta } { 2 } \right)$

Question 35

Use the sum-to-product formulas to select the sum or difference as a product. $\cos ( 5 \phi + 2 \pi ) + \cos 5 \phi$

A) $- 2 \cos 5 \phi \cos \pi$
B) $- 2 \cos 5 \pi$
C) $2 \cos ( 5 \phi + \pi ) \cos \pi$
D) $2 \cos ( \phi + \pi ) \cos \pi$
E) $2 \cos ( 5 \phi + \pi )$

Question 36

Convert the expression. $\sec \frac { b } { 2 }$

A) $\pm \sqrt { \frac { 2 \tan b } { \tan b + \sin b } }$
B) $\pm \sqrt { \frac { 2 \tan b } { \tan b + \csc b } }$
C) $\pm \sqrt { \frac { \tan b } { \tan b - \sin b } }$
D) $\pm \sqrt { \frac { \tan b } { \tan b + \sin b } }$
E) $\pm \sqrt { \frac { 2 \tan b } { \tan b - \sin b } }$

Question 37

Find the exact solutions of the given equation in the interval [0, 2 $\pi$ ). $\cos 2 x + 3 \cos x + 2 = 0$

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

Question 38

The mach number M of an airplane is the ratio of its speed to the speed of sound.When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure).The mach number is related to the apex angle $\theta$ of the cone by $\sin ( \theta / 5 ) = 1 / M$ .
Rewrite the equation in terms of $\theta$ .

A) $\theta = 5 \sin \left( \frac { 1 } { M } \right)$
B) $\theta = \sin \left( \frac { 1 } { M } \right)$
C) $\theta = \sin ^ { - 1 } \left( \frac { 1 } { M } \right)$
D) $\theta = 5 \sin ^ { - 1 } \left( \frac { 5 } { M } \right)$
E) $\theta = 5 \sin ^ { - 1 } \left( \frac { 1 } { M } \right)$

Question 39

Use the figure to find the exact value of the trigonometric function. $$\begin{array} { l } a = 8 , b = 9 \\c = 2 , d = 5\end{array}$$
Sin 2 $\alpha$

A) $\frac { 8 } { 145 }$
B) $\frac { 9 } { 145 }$
C) $\frac { 145 } { 2 }$
D) $\frac { 145 } { 144 }$
E) $\frac { 144 } { 145 }$

Question 40

Convert the expression. $7 \sec 2 \theta$

A) $\frac { 7 \sec ^ { 2 } \theta } { 2 + \sec ^ { 2 } \theta }$
B) $\frac { \sec ^ { 2 } \theta } { 2 - \sec ^ { 2 } \theta }$
C) $\frac { 7 \sec ^ { 2 } \theta } { 2 - \sec ^ { 2 } \theta }$
D) $\frac { 7 \sec ^ { 2 } \theta } { 2 - \cos ^ { 2 } \theta }$
E) $\frac { 7 \cos ^ { 2 } \theta } { 2 - \cos ^ { 2 } \theta }$

Question 41

The range of a projectile fired at an angle $\theta$ with the horizontal and with an initial velocity of v₀ feet per second is $r = \frac { 1 } { 32 } v _ { 0 } ^ { 2 } \sin 2 \theta$ where r is measured in feet.A golfer strikes a golf ball at 90 feet per second.Ignoring the effects of air resistance, at what angle must the golfer hit the ball so that it travels 150 feet? (Round your answer to the nearest degree.)

A)14°
B)36°
C)18°
D)27°
E)41°

Question 42

Use the figure below to find the exact value of the given trigonometric expression. $\cos \frac { \theta } { 2 }$

A) $\cos \frac { \theta } { 2 } = \frac { 7 \sqrt { 2 } } { 10 }$
B) $\cos \frac { \theta } { 2 } = \frac { 7 \sqrt { 2 } } { 7 }$
C) $\cos \frac { \theta } { 2 } = \frac { \sqrt { 2 } } { 10 }$
D) $\cos \frac { \theta } { 2 } = 7$
E) $\cos \frac { x } { 2 } = \frac { 7 \sqrt { 2 } } { 12 }$

Question 43

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) $\sin 50 ^ { \circ }$

Question 44

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 45

Find all solutions of the given equation in the interval [0, 2 $\pi$ ).
$\cos 3 x = \cos x$

A) $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$
B) $x = \frac { \pi } { 4 } , \frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$
C) $x = 0 , \frac { \pi } { 4 } , \frac { \pi } { 2 } , \pi$
D) $x = 0 , \frac { \pi } { 4 } , \pi , \frac { 3 \pi } { 4 }$
E) $x = \frac { \pi } { 8 } , \frac { 2 \pi } { 8 } , \frac { \pi } { 2 } , \frac { 3 \pi } { 8 } , \pi , \frac { 3 \pi } { 2 }$

Question 46

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 47

Use the sum-to-product formulas to find the exact value of the given expression.
$\cos 150 ^ { \circ } + \cos 30 ^ { \circ }$

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

Question 48

Use the half-angle formula to simplify the given expression.
$\sqrt { \frac { 1 + \cos 8 x } { 2 } }$

A) cos |2x|
B) cos |8x|
C) cos |16x|
D) cos |32x|
E) cos |4x|

Question 49

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 50

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 51

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 52

Find all solutions of the given equation in the interval [0, 2 $\pi$ ).
$\frac { 1 } { 2 } \sin 2 x = \cos x$

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

Question 53

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) $\cos 100 ^ { \circ }$

Question 54

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 55

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 56

Use the sum-to-product formulas to write the given expression as a product.
$\sin 5 \theta - \sin 3 \theta$

A) 2 sin 4 $\theta$ cos $\theta$
B) -2 sin 4 $\theta$ sin $\theta$
C)- 2 cos 4 $\theta$ cos $\theta$
D) 2 cos 4 $\theta$ sin $\theta$
E) 2 cos 4 $\theta$ cos $\theta$

Question 57

Use the half-angle formulas to determine the exact value of the given trigonometric expression.
$\tan \frac { 3 \pi } { 8 }$

A) $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } - 1$
B) $\tan \frac { 3 \pi } { 8 } = \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 }$
C) $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 + \sqrt { 2 } }$
D) $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } + 1$
E) $\tan \frac { 3 \pi } { 8 } = - \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 }$

Question 58

Use the product-to-sum formula to write the given product as a sum or difference. $12 \sin \frac { \pi } { 6 } \cos \frac { \pi } { 6 }$

A) $6 - 6 \cos \frac { \pi } { 12 }$
B) $6 \sin \frac { \pi } { 6 } + 6 \cos \frac { \pi } { 6 }$
C) $6 \sin \frac { \pi } { 3 } + 6 \sin 0$
D) $6 + 6 \cos \frac { \pi } { 12 }$
E) $- 6 \sin \frac { \pi } { 12 }$

Question 59

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 60

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 61

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 62

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 63

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 64

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

Question 65

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 66

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 67

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 68

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 69

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 70

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 71

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 72

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 73

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 74

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 75

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 76

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 77

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 78

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 79

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 80

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) $\sin ( 2 \theta + 1.0122 )$