Deck 35: Multiple Angle and Product to Sum Formulas

Question 1

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)​ $\left( \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 6 } \right)$

Answer 1

​ $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } + \sin \frac { \pi } { 6 } \right)$

Question 2

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)​ $2 \sin 9 \theta \cos 7 \theta$

Answer 2

​ $2 \sin 8 \theta \cos \theta$ ​

Question 3

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)​ $5 \left( \cos 65 ^ { \circ } + \sin 25 ^ { \circ } \right)$

Answer 3

$5 \left( \cos 25 ^ { \circ } + \cos 65 ^ { \circ } \right)$

Question 4

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 5

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 6

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 7

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 8

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 9

Use the figure to find the exact value of the trigonometric function. ​
Csc 2θ​ ​
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 10

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|

Question 11

Use the figure to find the exact value of the trigonometric function. ​
Cot 2θ​ ​
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 12

Use the figure to find the exact value of the trigonometric function. ​
Tan 2θ​ ​
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 13

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 14

Use the figure to find the exact value of the trigonometric function. ​
Cos 2θ​ ​
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 15

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 16

Use the figure to find the exact value of the trigonometric function. ​
Sec 2θ​ ​
A = 1,b = 8
​

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

Question 17

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 18

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 19

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 20

Use the figure to find the exact value of the trigonometric function. ​
Sin 2θ​ ​
A = 1,b = 2
​

A) $\frac { 5 } { 5 }$ ​
B) $\frac { 4 } { 5 }$ ​
C) $\frac { 5 } { 4 }$ ​

Question 21

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 22

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 23

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 24

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

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 25

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 26

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 θ of the cone by $$\sin ( \theta / 5 ) = 1 / M$$ .​ ​ Rewrite the equation in terms of θ.
​

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 27

Use the sum-to-product formulas to rewrite the sum or difference as a product. ​
Cos 3θ + cos 8θ
​

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 28

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 2α
​

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

Question 29

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 30

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 31

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 32

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 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 θ are related by​ $$\frac { x } { 2 } = 2 r \sin ^ { 2 } \frac { \theta } { 2 }$$ ​ Write a formula for x in terms of cos θ.​ ​

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

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 36

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 37

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 2β
​

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

Question 38

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 39

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

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 40

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 41

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 42

​Use the figure below to find the exact value of the given trigonometric expression. $$\sin \frac { \theta } { 2 }$$ ​​ 10 24
(figure not necessarily to scale)
​

A)​ $\frac { 5 } { 12 }$
B)​ $\frac { 5 } { 26 }$
C)​ $\frac { 5 \sqrt { 26 } } { 26 }$
D)​ $\frac { \sqrt { 26 } } { 26 }$
E)​ $\frac { \sqrt { 26 } } { 12 }$

Question 43

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 44

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 45

The range of a projectile fired at an angle θ 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 46

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

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 47

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

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 48

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 49

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 50

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

A)​​2 sin 4θ cos θ
B)​-2 sin 4θ sin θ
C)-​2 cos 4θ cos θ
D)​2 cos 4θ sin θ
E)​​2 cos 4θ cos θ