Deck 8: Graphs of the Trigonometric Functions

Question 1

Use the Pythagorean identities to simplify the expression. $$\frac { \cos ^ { 2 } t + \sin ^ { 2 } t } { \cot ^ { 2 } t + 1 }$$

A) $$\sin ^ { 2 } t$$
B) $$\csc ^ { 2 } \theta$$
C) $$\cos ^ { 2 } \theta$$
D) 1
E) $$\sec ^ { 2 } \theta$$

Answer 1

$$\sin ^ { 2 } t$$

Question 2

Refer to the graph of $$y = \cos x$$ in the figure. Specify the coordinates of the point I.

A) $$( - 3 \pi , 1 )$$
B) $$\left( - \frac { 3 \pi } { 2 } , 1 \right)$$
C) $$\left( - \frac { 7 \pi } { 2 } , 1 \right)$$
D) $$( - 5 \pi , 1 )$$
E) $$( - 2 \pi , 1 )$$

Answer 2

$$( - 2 \pi , 1 )$$

Question 3

Determine whether the equation for the graph has the form $$y = A \sin B x$$ or $$y = A \cos B x$$ ( with $$B > 0$$ ) and then find the values of $$A$$ and $${ B }$$ .

A) $$y = 5 \sin 6 x$$
B) $$y = 5 \sin \frac { x } { 6 }$$
C) $$y = - \frac { 1 } { 5 } \cos \frac { x } { 6 }$$
D) $$y = 5 \cos \frac { x } { 6 }$$
E) $$y = - 5 \cos 6 x$$

Answer 3

$$y = 5 \sin \frac { x } { 6 }$$

Question 4

State whether the function $$y = - \sin x$$ is increasing or decreasing on the interval. $$- \frac { 3 \pi } { 2 } < x < - \pi$$

A) increasing
B) decreasing

Question 5

State whether the function $$y = \sin x$$ is increasing or decreasing on the interval. $$- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$

A) decreasing
B) increasing

Question 6

Refer to the graph of $$y = - \sin x$$ in the figure. Specify the coordinates of the point I.

A) $$( 15.71,0 )$$
B) $$( 6.28,0 )$$
C) $$( 12.56,0 )$$
D) $$( 1.57,0 )$$
E) $$( 4.71,0 )$$

Question 7

A mass on a smooth tabletop is attached to a spring, as shown in the figure. The coordinate system has been chosen so that the equilibrium position of the mass corresponds to $$s = 0$$ . Assume that the simple harmonic motion is described by the equation $$s = 5 \cos \left( \frac { \pi t } { 3 } \right)$$ , where s is in centimeters and t is in seconds. Determine which table describes the s-coordinate of the mass at each of the following times: $$t =$$ 0 sec, 0.5 sec, 1 sec, and 2 sec. (One of these coordinates will involve a radical sign; for this case, use a calculator and round the final answer to two decimal places.)

A) $$\begin{array} { | c | c | } \hline \text { time (sec) } & \text { s-coordinate(cm) } \\\hline 0 & 5 \\\hline 0.5 & 4.33 \\\hline 1 & 2.5 \\\hline 2 & 2.5 \\\hline\end{array}$$
B) $$\begin{array} { | c | c | } \hline \text { time (sec) } & \text { s-coordinate(cm) } \\\hline 0 & - 2.5 \\\hline 0.5 & 4.33 \\\hline 1 & 2.5 \\\hline 2 & 5 \\\hline\end{array}$$
C) $$\begin{array} { | c | c | } \hline \text { time (sec) } & \text { s-coordinate(cm) } \\\hline 0 & 5 \\\hline 0.5 & 4.33 \\\hline 1 & 2.5 \\\hline 2 & - 2.5 \\\hline\end{array}$$
D) $$\begin{array} { | c | c | } \hline \text { time (sec) } & \text { s-coordinate(cm) } \\\hline 0 & 2.5 \\\hline 0.5 & 4.33 \\\hline 1 & 5 \\\hline 2 & - 2.5 \\\hline\end{array}$$
E) $$\begin{array} { | c | c | } \hline \text { time (sec) } & \text { s-coordinate(cm) } \\\hline 0 & 5 \\\hline 0.5 & 2.5 \\\hline 1 & 4.33 \\\hline 2 & - 2.5 \\\hline\end{array}$$

Question 8

Consider the function $$f ( x ) = \sin \left( x - \frac { \pi } { 4 } \right)$$ . Specify the phase shift.

A) $$\frac { \pi } { 4 }$$
B) $$\frac { \pi } { 8 }$$
C) $$- \frac { \pi } { 4 }$$
D) $$\frac { 3 \pi } { 4 }$$
E) $$0$$

Question 9

Consider the function $$y = \sin \frac { x } { 4 } - 5$$ on the interval $$[ 0,8 \pi ]$$ . Specify the intervals in which the function is increasing.

A) $$( 0,2 \pi )$$
B) $$( 0,2 \pi ) , ( 6 \pi , 8 \pi )$$
C) $$( 6 \pi , 8 \pi )$$
D) $$( 2 \pi , 6 \pi ) , ( 8 \pi , 9 \pi )$$
E) $$( 0,8 \pi )$$

Question 10

Consider the function $$y = \cos 3 x$$ . Specify the period.

A) $$\frac { \pi } { 3 }$$
B) $$\frac { 2 \pi } { 3 }$$
C) $$\frac { 6 \pi } { 7 }$$
D) $$2 \pi$$
E) $$- \frac { 2 \pi } { 3 }$$

Question 11

Consider the function $$y = 1 - \cos \frac { \pi x } { 4 }$$ on the interval $$[ 0,8 ]$$ . Determine the $$\chi$$ -intercepts by giving the $$\chi$$ -coordinate(s).

A) $$0,4$$
B) $$0,8$$
C) $$- 4$$
D) $$8$$
E) there are no $$\chi$$ -intercepts

Question 12

Given: $$\cot \theta + \tan \theta + 1$$ . Determine the right side of the identity equation.

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

Question 13

Consider the function $$y = 9 \cos \left( 8 x - \frac { \pi } { 3 } \right)$$ . Specify the amplitude.

A) $$18$$
B) $$3$$
C) $$\frac { 9 } { 8 }$$
D) $$9$$
E) $$\frac { 8 } { 9 }$$

Question 14

Consider the function $$y = 4 \sin x$$ . Specify the amplitude.

A) 1
B) 4
C) 5
D) 8
E) 2

Question 15

Consider the function $$y = \cos ( 6 x - \pi )$$ . Specify the period.

A) $$\frac { \pi } { 6 }$$
B) $$1$$
C) $$2 \pi$$
D) $$\frac { \pi } { 3 }$$
E) $$\frac { 4 \pi } { 3 }$$

Question 16

Compute $$\sec \alpha$$ , $$\cos \alpha$$ , and $$\sin \alpha$$ . $$\tan \alpha = \frac { 12 } { 5 }$$ and $$\cos \alpha > 0$$

A) $$\sec \alpha = - \frac { 12 } { 13 }$$ , $$\cos \alpha = \frac { 5 } { 13 }$$ , $$\sin \alpha = \frac { 12 } { 13 }$$
B) $$\sec \alpha = \frac { 13 } { 12 }$$ , $$\cos \alpha = \frac { 5 } { 13 }$$ , $$\sin \alpha = \frac { 12 } { 13 }$$
C) $$\sec \alpha = - \frac { 13 } { 5 }$$ , $$\cos \alpha = - \frac { 12 } { 13 }$$ , $$\sin \alpha = \frac { 5 } { 13 }$$
D) $$\sec \alpha = \frac { 5 } { 13 }$$ , $$\cos \alpha = \frac { 13 } { 5 }$$ , $$\sin \alpha = - \frac { 12 } { 13 }$$
E) $$\sec \alpha = \frac { 13 } { 5 }$$ , $$\cos \alpha = \frac { 5 } { 13 }$$ , $$\sin \alpha = \frac { 12 } { 13 }$$

Question 17

Use the Pythagorean identities to simplify the expression. $$\frac { \sec ^ { 2 } \theta - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta }$$

A) $$\sec ^ { 2 } \theta$$
B) $$\csc ^ { 2 } \theta$$
C) $$\cos ^ { 2 } \theta$$
D) $$\sin ^ { 2 } \theta$$
E) 1

Question 18

Consider the function $$y = 6 + \sin 5 x$$ . Specify the period.

A) $$- \frac { 2 } { 5 } \pi$$
B) $$\frac { 1 } { 6 } \pi$$
C) $$\frac { 1 } { 5 } \pi$$
D) $$\frac { 1 } { 5 }$$
E) $$\frac { 2 } { 5 } \pi$$

Question 19

Specify the period and amplitude for the function.

A) period: 8; amplitude: 8
B) period: 4; amplitude: 16
C) period: 2; amplitude: 16
D) period: 4; amplitude: 8
E) period: 2; amplitude: 8

Question 20

Consider the function $$y = 8 \sin \frac { \pi x } { 8 }$$ on the interval $$[ 0,16 ]$$ . Determine the $$\chi$$ -intercepts by giving the $$\chi$$ -coordinate(s).

A) $$0,8,24$$
B) $$0,8,16$$
C) $$8,16$$
D) $$0,16$$
E) $$0,8 \pi , 16 \pi$$

Question 21

Graph the function for one period, and show (or specify) intercepts and asymptotes. $$y = \cot \left( x + \frac { \pi } { 3 } \right)$$

A)x-intercept: none; y-intercept:
$$\frac { 2 \sqrt { 3 } } { 3 }$$ ; asymptotes: $x = - \frac { \pi } { 3 }$ and $$x = \frac { 2 \pi } { 3 }$$

B) x-intercept: none; y-intercept:
$$- \frac { 2 \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$
C) x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$\frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$
D) x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$\frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$
E) x-intercept: $$\frac { \pi } { 6 }$$ ; y-intercept: $$- \frac { \sqrt { 3 } } { 3 }$$ ; asymptotes: $$x = - \frac { \pi } { 3 }$$ and $$x = \frac { 2 \pi } { 3 }$$

Question 22

Graph the function for one period, and show (or specify) intercepts and asymptotes. $$y = - \frac { 1 } { 4 } \csc ( 4 \pi x )$$

A) x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$
B) x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$
C) x-intercept: none; y-intercept: none; asymptotes: $$x = 0$$ and $$x = \frac { 1 } { 4 }$$
Chapter 8
Answer Section
MULTIPLE CHOICE
D) x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$
E) x-intercept: none; y-intercept: none; asymptotes: $$x = - \frac { 1 } { 4 }$$ , $$x = 0$$ and $$x = \frac { 1 } { 4 }$$

Question 23

Graph the function for one period, and show (or specify) intercepts and asymptotes. $$y = - \csc \left( \frac { x } { 4 } \right)$$

A) x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$
B) x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ and $$x = 0$$
C) x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$
D) x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$
E) x-intercept: none; y-intercept: none; asymptotes: $$x = - 4 \pi$$ , $$x = 0$$ and $$x = 4 \pi$$

Question 24

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $$y = - \tan \left( x - \frac { \pi } { 6 } \right)$$

A)
B)
C)
D)
E)

Question 25

Graph the function for one period, and show (or specify) the intercepts and the asymptotes. $$y = \tan \left( x + \frac { \pi } { 2 } \right)$$

A)
B)
C)
D)
E)