Deck 33: Solving Trigonometric Equations

Question 1

Solve the following equation.​ $2 \sin ^ { 2 } x = 6 \cos ^ { 2 } x$ ​

A)​ $\frac { \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
B)​ $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
C)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$
D)​ $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
E)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$

Answer 1

​ $\frac { \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$

Question 2

Find all solutions of the following equation in the interval [0,2π).​ $3 \sin x + \csc x = 0$ ​

A) $\frac { 5 \pi } { 2 } + 2 \pi , \frac { 5 \pi } { 2 } + 2 \pi$
B)No solution
C)​ $\frac { \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
D)​ $\frac { 3 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
E)​ $\frac { 7 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$

Answer 2

No solution

Question 3

Solve the following equation.​ $\tan 3 x ( 8 \tan x - 8 ) = 0$ ​

A) $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
B)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$
C)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$
D)​ $\frac { n \pi } { 3 } , \frac { \pi } { 4 } + n \pi$
E)​ $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$

Answer 3

​ $\frac { n \pi } { 3 } , \frac { \pi } { 4 } + n \pi$

Question 4

Solve the following equation.​ $12 \sec ^ { 2 } x - 16 = 0$ ​

A) $\frac { 3 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
B)​ $\frac { 7 \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$
C)​ $\frac { 5 \pi } { 2 } + 2 \pi , \frac { 5 \pi } { 2 } + 2 \pi$
D)​ $\frac { \pi } { 6 } + n \pi , \frac { 5 \pi } { 6 } + n \pi$
E)​ $\frac { \pi } { 2 } + 2 \pi , \frac { 3 \pi } { 2 } + 2 \pi$

Question 5

Solve the following equation.​ $$4 \cos x + 2 = 0$$ ​

A)​ $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
B)​ $$\frac { 2 \pi } { 3 } + 2 n \pi , \frac { 4 \pi } { 3 } + 2 n \pi$$
C)​ $$\frac { 7 \pi } { 6 } + 2 n \pi , \frac { 11 \pi } { 6 } + 2 n \pi$$
D)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$
E)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$

Question 6

Find all solutions of the following equation in the interval [0,2π).​ $3 \sec x - 3 \tan x = 3$ ​

A) $\frac { 7 \pi } { 2 } , 2 \pi , \frac { 3 \pi } { 2 }$
B)0
C)​ $\frac { \pi } { 2 } , 2 \pi , \frac { 3 \pi } { 2 }$
D)​ $\frac { 3 \pi } { 2 } , 2 \pi , \frac { 3 \pi } { 4 }$
E)​ $\frac { 5 \pi } { 2 } , 2 \pi$

Question 7

Find all solutions of the following equation in the interval [0,2π).​ $4 \tan x + 4 \sqrt { 3 } = 0$ ​

A) $\frac { 5 \pi } { 3 }$
B)​ $\frac { \pi } { 3 }$
C)​ $\frac { 3 \pi } { 4 }$
D)​ $\frac { 2 \pi } { 3 }$
E)​ $\frac { 2 \pi } { 5 }$

Question 8

Solve the following equation.​ $\left( 3 \tan ^ { 2 } x - 3 \right) \left( \tan ^ { 2 } x - 3 \right) = 0$ ​

A)​ $\frac { \pi } { 4 } + n \pi , \frac { \pi } { 2 } + n \pi , \frac { \pi } { 6 } + n \pi , \frac { 5 \pi } { 6 } + n \pi$
B)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi , \frac { \pi } { 6 } + n \pi , \frac { 5 \pi } { 6 } + n \pi$
C)​ $\frac { 5 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi , \frac { \pi } { 6 } + n \pi , \frac { 5 \pi } { 6 } + n \pi$
D)​ $\frac { \pi } { 4 } + n \pi , \frac { 5 \pi } { 4 } + n \pi , \frac { \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
E)​ $\frac { \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi , \frac { \pi } { 6 } + n \pi , \frac { 5 \pi } { 6 } + n \pi$

Question 9

Find all solutions of the following equation in the interval [0,2π).​ $5 \sec x \csc x = 10 \csc x$ ​

A) $\frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$
B)​ $\frac { 5 \pi } { 2 } , \frac { 5 \pi } { 3 }$
C)​ $\frac { 7 \pi } { 2 } , \frac { \pi } { 3 }$
D)​ $\frac { 3 \pi } { 2 } , \frac { \pi } { 3 }$
E)​ $2 \pi , \pi$

Question 10

Solve the following equation.​ $2 \tan ^ { 2 } 3 x = 6$ ​

A) $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
B)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$
C)​ $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
D)​ $\frac { \pi } { 9 } + \frac { n \pi } { 3 } , \frac { 2 \pi } { 9 } + \frac { n \pi } { 3 }$
E)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$

Question 11

Solve the following equation.​ $\cos 2 x ( 4 \cos x + 2 ) = 0$ ​

A) $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi , \frac { 4 \pi } { 3 } + 2 n \pi$
B)​ $\frac { \pi } { 4 } + \frac { n \pi } { 2 } , \frac { 2 \pi } { 3 } + 2 n \pi , \frac { 4 \pi } { 3 } + 2 n \pi$
C)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$
D)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$
E)​ $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$

Question 12

Solve the following equation. ​​ $$10 \sin x + 5 = 0$$ ​

A)​ $\frac { 7 \pi } { 6 } + 2 n \pi , \frac { 11 \pi } { 6 } + 2 n \pi$
B)​ $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
C)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$
D)​ $$\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$$
E)​ $$\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$$

Question 13

Solve the following equation.​ $9 \cot ^ { 2 } x - 3 = 0$ ​

A)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$
B)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$
C)​ $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
D)​ $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
E) $\frac { \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$

Question 14

Solve the following equation.​ $\sin x ( 5 \sin x + 5 ) = 0$ ​

A)​ $n \pi , \frac { \pi } { 2 } + 2 n \pi$
B)​ $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
C) $n \pi , \frac { 3 \pi } { 2 } + 2 n \pi$
D)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$
E)​ $\pi , \frac { 5 \pi } { 2 } + 2 n \pi$

Question 15

Find all solutions of the following equation in the interval [0,2π).​ $5 \sec ^ { 2 } x - 5 = 0$ ​

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

Question 16

Solve the following equation.​ $8 \sin ^ { 2 } 2 x = 4$ ​

A) $\frac { 4 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
B)​ $\frac { \pi } { 2 } + n \pi , \frac { 2 \pi } { 5 } + n \pi$
C)​ $\frac { \pi } { 8 } + \frac { n \pi } { 2 } , \frac { 3 \pi } { 8 } + \frac { n \pi } { 2 }$
D)​ $\frac { 2 \pi } { 3 } + n \pi , \frac { 2 \pi } { 3 } + n \pi$
E)​ $\frac { \pi } { 3 } + n \pi , \frac { \pi } { 3 } + n \pi$

Question 17

Find all solutions of the following equation in the interval [0,2π).​ $15 \tan ^ { 3 } x = 5 \tan x$ ​

A)​ $0 , \pi , \frac { \pi } { 6 } , \frac { n \pi } { 6 } , \frac { 2 n \pi } { 6 } , \frac { 3 n \pi } { 6 }$
B)​ $0 , \pi , \frac { \pi } { 6 } , \frac { 3 \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 }$
C)​ $0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$
D)​ $0 , \pi , \frac { \pi } { 6 } , \frac { 2 \pi } { 6 } , \frac { 3 \pi } { 6 } , \frac { 11 \pi } { 6 }$
E)​ $0 , \pi , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { \pi } { 6 } , \frac { 13 \pi } { 6 }$

Question 18

Solve the following equation.​ $$5 \sqrt { 3 } \csc x - 10 = 0$$ ​

A) $\frac { \pi } { 3 } + 2 n \pi , \frac { 2 \pi } { 3 } + 2 n \pi$
B)​ $\frac { \pi } { 2 } + 2 n \pi , \frac { 2 \pi } { 2 } + 2 n \pi$
C)​ $\frac { \pi } { 3 } + 2 n \pi , \frac { 2 \pi } { 3 } + \pi$
D)​ $\frac { \pi } { 3 } + \pi , \frac { 2 \pi } { 3 } + 2 n \pi$
E)​ $\frac { \pi } { 4 } + 2 n \pi , \frac { \pi } { 2 } + 2 n \pi$

Question 19

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

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

Question 20

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

A) $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$
B)​ $\frac { \pi } { 2 } , \frac { 5 \pi } { 2 }$
C)​ $\frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 }$
D)​ $\frac{\pi}{2}, \frac{5 \pi}{2}$
E)​ $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$

Question 21

Solve the following equation. ​
Secx - 2 = 0
​

A)​ $x = \frac { \pi } { 5 } + 2 n \pi$ and $x = \frac { 7 \pi } { 5 } + 2 n \pi$ ,where n is a integer
B)​ $x = \frac { \pi } { 3 } + 2 n \pi$ and $x = \frac { 5 \pi } { 3 } + 2 n \pi$ ,where n is a integer
C)​ $x = \frac { \pi } { 6 } + 2 n \pi$ and $x = \frac { 7 \pi } { 6 } + 2 n \pi$ ,where n is a integer
D)​ $x = \frac { \pi } { 6 } + 2 n \pi$ and $x = \frac { 5 \pi } { 6 } + 2 n \pi$ ,where n is a integer
E)​ $x = \frac { \pi } { 4 } + 2 n \pi$ and $x = \frac { 5 \pi } { 4 } + 2 n \pi$ ,where n is a integer

Question 22

Use a graphing utility to graph the function.​ $$f ( x ) = 1.1 ( 2 \sin x + \cos 2 x )$$ ​

A)​
B)​
C)​
D)​
E)​

Question 23

Solve the multiple-angle equation.​ $2 \sec 4 x = 4$ ​

A) $\frac { \pi } { 12 } - \frac { n \pi } { 3 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$
B)​ $\frac { 3 \pi } { 4 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
C)​ $\frac { \pi } { 2 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
D)​ $\frac { 5 \pi } { 12 } + \frac { n \pi } { 3 } , \frac { \pi } { 12 } + \frac { n \pi } { 2 }$
E)​ $\frac { \pi } { 12 } + \frac { n \pi } { 2 } , \frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$

Question 24

Use inverse functions where needed to find all solutions of the equation in the interval [0,2π).​ $\sec ^ { 2 } x - 6 \sec x = 0$ ​

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

Question 25

Solve the multiple-angle equation.​ $4 \tan 3 x = 4$ ​

A) $\frac { 5 \pi } { 12 } + \frac { n \pi } { 2 }$
B)​ $\frac { \pi } { 12 } - \frac { \pi } { 2 }$
C)​ $\frac { \pi } { 3 } + \frac { n \pi } { 3 }$
D)​ $\frac { \pi } { 12 } + \frac { n \pi } { 3 }$
E)​ $\frac { 3 \pi } { 4 } + \frac { n \pi } { 3 }$

Question 26

Use a graphing utility to graph the function.​ $$f ( x ) = \sin x - \cos x$$ ​

A)​
B)​
C)​
D)​
E)​

Question 27

Solve the multiple-angle equation.​ $\sin \frac { 3 x } { 6 } = - \frac { \sqrt { 3 } } { 2 }$ ​

A) $\frac { \pi } { 3 } + 3 n \pi , \frac { 2 \pi } { 3 } + 3 n \pi$
B)​ $\frac { \pi } { 3 } + 2 n \pi , \frac { 2 \pi } { 3 } + 2 n \pi$
C)​ $\frac { 8 \pi } { 3 } + 4 n \pi , \frac { 10 \pi } { 3 } + 4 n \pi$
D)​ $\frac { 5 \pi } { 3 } + 2 n \pi , \frac { 4 \pi } { 3 } + 3 n \pi$
E)​ $\frac { \pi } { 3 } + 4 n \pi , \frac { 2 \pi } { 3 } + 4 n \pi$

Question 28

Use inverse functions where needed to find all solutions of the equation in the interval [0,2π).​ $\cot ^ { 2 } x - 25 = 0$ ​

A) $\arctan \left( \frac { 1 } { 5 } \right) , \arctan \left( \frac { 1 } { 5 } \right) + \pi , \arctan \left( \frac { 1 } { 5 } \right) + \pi , \arctan \left( \frac { 1 } { 5 } \right) + 2 \pi$
B)​ $\arctan \left( \frac { 1 } { 5 } \right) , \arctan \left( \frac { 1 } { 5 } \right) + \pi , \arctan \left( \frac { 1 } { 5 } \right) + \pi , \arctan \left( \frac { 1 } { 5 } \right) + \pi$
C)​ $\arctan \left( \frac { 1 } { 5 } \right) , \arctan \left( \frac { 1 } { 5 } \right) + \pi , \arctan \left( - \frac { 1 } { 5 } \right) + \pi , \arctan \left( - \frac { 1 } { 5 } \right) + 2 \pi$
D)​ $\arctan \left( \frac { 1 } { 5 } \right) , \arctan \left( \frac { 1 } { 5 } \right) + 2 \pi , \arctan \left( - \frac { 1 } { 5 } \right) + 2 \pi , \arctan \left( - \frac { 1 } { 5 } \right) + 2 \pi$
E)​ $\arctan \left( \frac { 1 } { 5 } \right) , \arctan \left( \frac { 1 } { 5 } \right) + 2 \pi , \arctan \left( - \frac { 1 } { 5 } \right) + \pi , \arctan \left( - \frac { 1 } { 5 } \right) + 2 \pi$

Question 29

A weight is oscillating on the end of a spring (see figure).The position of the weight relative to the point of equilibrium is given by $y = \frac { 1 } { 12 } ( \cos 7 t - 3 \sin 7 t )$ ,where y is the displacement (in meters)and t is the time (in seconds).Find the times when the weight is at the point of equilibrium (y = 0)for $0 \leq t \leq 1$ .​ ​ ​

A)0.05sec,0.49sec,0.94sec
B)0.05sec,0.05sec,0.94sec
C)-0.40sec,0.49sec,0.94sec
D)-0.40sec,0.49sec,-0.40sec
E)0.05sec,-0.40sec,0.94sec

Question 30

The table shows the average daily high temperatures in Houston H (in degrees Fahrenheit)for month t,with t = 1 corresponding to January. $$\begin{array} { | c | c | } \hline \text { Month, } t & \text { Houston, } H \\\hline 1 & 62.3 \\\hline 2 & 66.3 \\\hline 3 & 73.3 \\\hline 4 & 79.3 \\\hline 5 & 85.3 \\\hline 6 & 90.3 \\\hline 7 & 93.3 \\\hline 8 & 93.4 \\\hline 9 & 89.3 \\\hline 10 & 82.3 \\\hline 11 & 72.3 \\\hline 12 & 64.3 \\\hline\end{array}$$
Select the correct scatter plot from the above data.

A)​
B)​
C)​
D)​
E)​

Question 31

Solve the following equation. $\tan ^ { 4 } x - 1 = 0$

A)​ $x = \frac { \pi } { 2 } + n \pi$ ,where n is an integer
B) $x = n \pi \text { and } x = \frac { 3 \pi } { 2 } + 2 n \pi$ ,where n is an integer
C)​ $x = n \pi \text { and } x = \frac { 3 \pi } { 4 } + n \pi$ ,where n is an integer
D) $x = \frac { \pi } { 4 } + \frac { n \pi } { 2 }$ ,where n is an integer
E)​ $x = n \pi \text { and } x = \frac { \pi } { 2 } + n \pi$ ,where n is an integer

Question 32

Solve the following equation.​ $\cos ^ { 2 } x + \cos x = 0$ ​

A)​ $x = \frac { 3 \pi } { 4 } + n \pi \text { and } x = \frac { 5 \pi } { 4 } + n \pi$ ,where n is an integer
B)​ $x = \pi + 2 n \pi \text { and } x = \frac { \pi } { 2 } + n \pi$ ,where n is an integer
C)​ $x = \frac { 3 \pi } { 4 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 4 } + 2 n \pi$ ,where n is an integer
D)​ $x = n \pi \text { and } x = \frac { 3 \pi } { 2 } + n \pi$ ,where n is an integer
E)​ $x = \pi + n \pi \text { and } x = \frac { \pi } { 2 } + 2 n \pi$ ,where n is an integer

Question 33

Use a graphing utility to graph the function.​ $$f ( x ) = 2 ( \sin x \cos x )$$ ​

A)​
B)​
C)​
D)​
E)​

Question 34

Use a graphing utility to graph the function. ​​ $$f ( x ) = \left( \sin ^ { 2 } x - \cos x \right)$$ ​

A)​
B)​
C)​
D)​
E)​

Question 35

Solve the multiple-angle equation.​ $\cos \frac { 3 x } { 6 } = \frac { \sqrt { 2 } } { 2 }$ ​

A) $\frac { \pi } { 3 } + 2 n \pi , \frac { 2 \pi } { 3 } + 2 n \pi$
B)​ $\frac { \pi } { 3 } + 4 n \pi , \frac { 2 \pi } { 3 } + 4 n \pi$
C)​ $\frac { \pi } { 3 } + 3 n \pi , \frac { 2 \pi } { 3 } + 3 n \pi$
D)​ $\frac { \pi } { 2 } + 4 n \pi , \frac { 7 \pi } { 2 } + 4 n \pi$
E)​ $\frac { 5 \pi } { 3 } + 2 n \pi , \frac { 4 \pi } { 3 } + 3 n \pi$

Question 36

Use inverse functions where needed to find all solutions of the equation in the interval [0,2π).​ $\csc ^ { 2 } x - 7 \csc x = 0$ ​

A)​ $\arcsin \left( - \frac { 1 } { 7 } \right) + \pi , \arcsin \left( - \frac { 1 } { 7 } \right)$
B)​ $\arcsin \left( - \frac { 1 } { 7 } \right) + \pi , \arcsin \left( \frac { 1 } { 7 } \right)$
C)​ $\arcsin \left( - \frac { 1 } { 7 } \right) + \pi , \arcsin \left( \frac { 1 } { 7 } \right) + 2 \pi$
D)​ $\arcsin \left( - \frac { 1 } { 7 } \right) + 2 \pi , \arcsin \left( \frac { 1 } { 7 } \right)$
E)​ $\arcsin \left( \frac { 1 } { 7 } \right) + \pi , \arcsin \left( \frac { 1 } { 7 } \right)$

Question 37

Which of the following is a solution to the given equation? ​
Cscx + 2 = 0
​

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

Question 38

Which of the following is a solution to the given equation?​ $2 \cos x - 1 = 0$ ​

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

Question 39

Use a graphing utility to graph the function.​ $f ( x ) = 5 \left( \sin ^ { 2 } x + \cos x \right)$ ​

A)​
B)​
C)​
D)​
E)​

Question 40

Determine whether the statement is true or false.Justify your answer. ​
The equation $2 \sin 4 t - 1 = 0$ has four times the number of solutions in the interval [0,2π)as the equation $2 \sin t - 1 = 0$ .
​

A)False
B)True

Question 41

Solve the following equation.​ $2 \sin x - 1 = 0$

A)​ $x = \frac { 2 \pi } { 3 } + 2 n \pi \text { and } x = \frac { 4 \pi } { 3 } + 2 n \pi$ ,where n is an integer
B)​ $x = \frac { \pi } { 6 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 6 } + 2 n \pi$ ,where n is an integer
C)​ $x = \frac { \pi } { 3 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 3 } + 2 n \pi$ ,where n is an integer
D)​ $x = \frac { \pi } { 4 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 4 } + 2 n \pi$ ,where n is an integer

Question 42

Using the grid provided,sketch the graph of the given function in the interval (-5,5)and then determine the x-intercepts,if any. $$y = \sin \left( \frac { \pi x } { 2 } \right) - 1$$

A)(-3,0)and (1,0)
B)(-2,0)
C)(-2,0)and (2,0)
D)no x-intercepts
E)(-4,0), (0,0),and (4,0)

Question 43

Which of the following is a solution to the given equation? ​
Cscx - 2 = 0

A)​ $x = \frac { 2 \pi } { 3 }$
B)​ $x = \frac { \pi } { 4 }$
C)​ $x = \frac { \pi } { 4 }$ .
D)​ $x = \frac { \pi } { 6 }$
E)​ $x = \frac { 7 \pi } { 4 }$

Question 44

A Ferris wheel is built such that the height h (in feet)above the ground of a seat on the wheel at time t (in seconds)can be modeled by $h ( t ) = 67 + 54 \sin \left( \frac { \pi } { 18 } t - \frac { \pi } { 2 } \right)$ .The wheel makes one revolution every 36 seconds and the ride begins when t = 0.During the first 36 seconds of the ride,when will a person,who starts at the bottom of the Ferris wheel,be 67 feet above the ground? ​

A)9 seconds and 22 seconds
B)9 seconds and 27 seconds
C)10 seconds and 22 seconds
D)9 seconds and 17 seconds
E)10 seconds and 17 seconds

Question 45

Use the Quadratic Formula to solve the given equation on the interval [ $0 , \frac { \pi } { 2 }$ );then use a graphing utility to approximate the angle x.Round answers to three decimal places.​ $75 \cos ^ { 2 } x - 34 \cos x + 3 = 0$ ​

A)x = 1.229,1.433
B)​x = 1.227,1.422
C)​x = 1.233,1.469
D)​x = 1.231,1.451
E)​x = 1.235,1.480

Question 46

Solve the multiple-angle equation. $\tan \frac { x } { 2 } = \frac { \sqrt { 3 } } { 3 }$

A)​ $x = \frac { 5 \pi } { 3 } + n \pi$
B)​ $x = \frac { 7 \pi } { 6 } + n \pi \text { and } \frac { 11 \pi } { 6 } + n \pi$
C)​ $x = \frac { 2 \pi } { 3 } + n \pi$
D)​ $x = \frac { 5 \pi } { 6 } + n \pi \text { and } \frac { 7 \pi } { 6 } + n \pi$
E) $x = \frac { \pi } { 3 } + 2 n \pi$

Question 47

Use a graphing utility to approximate the solutions (to three decimal places)of the given equation in the interval [0,2π).​ $$\sin 2 x + 1.5 \cos x = 0$$ ​

A)x = 1.624,1.932,5.776,5.997
B)​x = 1.101,2.118,3.982,5.104
C)​x = 1.571,3.990,4.712,5.435
D)​x = 1.055,3.785,4.652,5.721
E)​x = 1.484,3.799,4.626,5.490

Question 48

The monthly sales S (in hundreds of units)of baseball equipment for an Internet sporting goods site are approximated by $S = 55.7 - 37.5 \cos \frac { \pi t } { 6 }$ where t is the time (in months),with t = 1 corresponding to January.Determine the months when sales exceed 7700 units at any time during the month.

A)April through August
B)July through May
C)June through August
D)April through July
E)May through July

Question 49

Use a graphing utility to approximate the solutions (to three decimal places)of the given equation in the interval $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ . ​​ $6 \sin 2 x - 8 \cos x + 9 \sin x = 6$ ​

A)​x = 0.398
B)​​x = 1.336
C)​​x = 0.094
D)​​x = 0.73
E)​​x = 0.139

Question 50

Use inverse functions where needed to find all solutions (if they exist)of the given equation on the interval [0,2π).​ $2 \cos ^ { 2 } x - \cos x - 1 = 0$ ​

A)​ $x = \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$
B) $x = 0 , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 }$
C) $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$
D) $x = \frac { \pi } { 3 } , \pi , \frac { 5 \pi } { 3 }$
E)​solution does not exist

Question 51

Find all solutions of the following equation in the interval [0,2π).​ $\sin ^ { 3 } x = \sin x$ ​

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

Question 52

Solve the multiple-angle equation.​ $\sin \frac { x } { 2 } = \frac { \sqrt { 2 } } { 2 }$

A)​​ $x = \frac { 2 \pi } { 3 } + 2 n \pi \text { and } x = \frac { 5 \pi } { 3 } + 2 n \pi$ ,where n is an integer
B)​ $x = \frac { 7 \pi } { 6 } + 2 n \pi \text { and } x = \frac { 11 \pi } { 6 } + 2 n \pi$ ,where n is an integer
C)​ $x = \frac { \pi } { 2 } + n \pi$ ,where n is an integer
D) $x = \frac { \pi } { 2 } + 2 n \pi \text { and } x = \frac { 3 \pi } { 2 } + 2 n \pi$ ,where n is an integer
E) $x = \frac { \pi } { 2 } + 4 n \pi \text { and } x = \frac { 3 \pi } { 2 } + 4 n \pi$ ,where n is an integer

Question 53

The horizontal distance d (in feet)traveled by a projectile with an initial speed of v feet per second is modeled by $d = \frac { v ^ { 2 } } { 32 } \sin 2 \theta$ , where θ is the angle at which the projectile is launched.
Find the horizontal distance traveled by a golf ball that is hit with an initial speed of 80 feet per second when the ball is hit at an angle of $\theta = 60 ^ { \circ }$ .Round to the nearest foot.

A)260
B)346
C)173
D)303
E)87

Question 54

Which of the following is a solution to the given equation? $2 \cos x - 1 = 0$

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