Content check — Precalculus, Chapter 6: Periodic Functions

Try It #1 §6.1 source ↗

Determine the period of the function

g (x) =

\cos (\frac{x}{3}) .

Answer

6 π

Try It #2 §6.1 source ↗

What is the amplitude of the sinusoidal function

f (x) =

\frac{1}{2} \sin (x) ?

Is the function stretched or compressed vertically?

Answer

\frac{1}{2}

compressed

Try It #3 §6.1 source ↗

Determine the direction and magnitude of the phase shift for

f (x) =

3 \cos (x - \frac{π}{2}) .

Answer

\frac{π}{2};

right

Try It #4 §6.1 source ↗

Determine the direction and magnitude of the vertical shift for

f (x) =

3 \sin (x) +

2.

Answer

2 units up

Try It #5 §6.1 source ↗

Determine the midline, amplitude, period, and phase shift of the function

y =

\frac{1}{2} \cos (\frac{x}{3} - \frac{π}{3}) .

Answer

midline:

y = 0;

amplitude:

| A | = \frac{1}{2};

period:

P = \frac{2 π}{| B |} = 6 π;

phase shift:

\frac{C}{B} = π

Try It #6 §6.1 source ↗

Determine the formula for the sine function in Figure 16.

A graph of sin(x)+2. Period of 2pi, amplitude of 1, and range of [1, 3].

Figure 16

Answer

f (x) =

\sin (x) +

2

Try It #7 §6.1 source ↗

Write a formula for the function graphed in Figure 18.

A graph of 4sin((pi/5)x-pi/5)+4. Graph has period of 10, amplitude of 4, range of [0,8].

Figure 18

Answer

two possibilities:

y =

4 \sin (\frac{π}{5} x - \frac{π}{5}) +

4

or

y =

-

4 \sin (\frac{π}{5} x + \frac{4 π}{5}) +

4

Try It #8 §6.1 source ↗

Sketch a graph of

g (x) =

-

0.8 \cos (2 x) .

Determine the midline, amplitude, period, and phase shift.

Answer

A graph of -0.8cos(2x). Graph has range of [-0.8, 0.8], period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x).

midline: $y = 0;$ amplitude: $| A | = 0.8;$ period: $P = \frac{2 π}{| B |} = π;$ phase shift: $\frac{C}{B} = 0$ or none

Try It #9 §6.1 source ↗

Draw a graph of

g (x) =

-

2 \cos (\frac{π}{3} x + \frac{π}{6}) .

Determine the midline, amplitude, period, and phase shift.

Answer

A graph of -2cos((pi/3)x+(pi/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left.

midline: $y = 0;$ amplitude: $| A | = 2;$ period: $P = \frac{2 π}{| B |} = 6;$ phase shift: $\frac{C}{B} = - \frac{1}{2}$

Try It #10 §6.1 source ↗

What is the amplitude of the function

f (x) =

7 \cos (x) ?

Sketch a graph of this function.

Answer

7

A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].

Try It #11 §6.1 source ↗

A weight is attached to a spring that is then hung from a board, as shown in Figure 25. As the spring oscillates up and down, the position $y$ of the weight relative to the board ranges from $–1$ in. (at time $x = 0)$ to $–7$ in. (at time $x = π)$ below the board. Assume the position of $y$ is given as a sinusoidal function of $x .$ Sketch a graph of the function, and then find a cosine function that gives the position $y$ in terms of $x .$

An illustration of a spring with length y.

Figure 25

Answer

$y = 3 \cos (x) - 4$

A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).

6.1-1 §6.1 source ↗

Why are the sine and cosine functions called periodic functions?

Answer

The sine and cosine functions have the property that

f (x + P) = f (x)

for a certain

P .

This means that the function values repeat for every

P

units on the x-axis.

6.1-3 §6.1 source ↗

For the equation

A \cos (B x + C) +

D,

what constants affect the range of the function and how do they affect the range?

Answer

The absolute value of the constant

A

(amplitude) increases the total range and the constant

D

(vertical shift) shifts the graph vertically.

6.1-5 §6.1 source ↗

How can the unit circle be used to construct the graph of

f (t) = \sin t ?

Answer

At the point where the terminal side of

t

intersects the unit circle, you can determine that the

\sin t

equals the y-coordinate of the point.

6.1-7 §6.1 source ↗

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ Round answers to two decimal places if necessary.

$f (x) = \frac{2}{3} \cos x$

Answer

A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3].

amplitude: $\frac{2}{3};$ period: $2 π;$ midline: $y = 0;$ maximum: $y = \frac{2}{3}$ occurs at $x = 2 π;$ minimum: $y = - \frac{2}{3}$ occurs at $x = π;$ for one period, the graph starts at 0 and ends at $2 π$

6.1-9 §6.1 source ↗

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ Round answers to two decimal places if necessary.

$f (x) = 4 \sin x$

Answer

A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].

amplitude: 4; period: $2 π;$ midline: $y = 0;$ maximum $y = 4$ occurs at $x = \frac{π}{2};$ minimum: $y = - 4$ occurs at $x = \frac{3 π}{2};$ one full period occurs from $x = 0$ to $x = 2 π$

6.1-11 §6.1 source ↗

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ Round answers to two decimal places if necessary.

$f (x) =$ $\cos (2 x)$

Answer

A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].

amplitude: 1; period: $π;$ midline: $y = 0;$ maximum: $y = 1$ occurs at $x = π;$ minimum: $y = - 1$ occurs at $x = \frac{π}{2};$ one full period is graphed from $x = 0$ to $x = π$

6.1-13 §6.1 source ↗

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ Round answers to two decimal places if necessary.

$f (x) =$ $4 \cos (π x)$

Answer

A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].

amplitude: 4; period: 2; midline: $y = 0;$ maximum: $y = 4$ occurs at $x = 2;$ minimum: $y = - 4$ occurs at $x = 1$

6.1-15 §6.1 source ↗

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ Round answers to two decimal places if necessary.

$y =$ $3 \sin (8 (x + 4)) +$ $5$

Answer

A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4.

amplitude: 3; period: $\frac{π}{4};$ midline: $y = 5;$ maximum: $y = 8$ occurs at $x = 0.12;$ minimum: $y = 2$ occurs at $x = 0.516;$ horizontal shift: $- 4;$ vertical translation 5; one period occurs from $x = 0$ to $x = \frac{π}{4}$

6.1-17 §6.1 source ↗

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ Round answers to two decimal places if necessary.

$y =$ $5 \sin (5 x + 20) -$ $2$

Answer

A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3].

amplitude: 5; period: $\frac{2 π}{5};$ midline: $y = −2;$ maximum: $y = 3$ occurs at $x = 0.08;$ minimum: $y = −7$ occurs at $x = 0.71;$ phase shift: $−4;$ vertical translation: $−2;$ one full period can be graphed on $x = 0$ to $x = \frac{2 π}{5}$

6.1-19 §6.1 source ↗

For the following exercises, graph one full period of each function, starting at $x = 0.$ For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

$f (t) =$ $-$ $\cos (t + \frac{π}{3}) +$ $1$

Answer

A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left.

amplitude: 1 ; period: $2 π;$ midline: $y = 1;$ maximum: $y = 2$ occurs at $x = 2.09;$ minimum: $y = 0$ occurs at $t = 5.24;$ phase shift: $- \frac{π}{3};$ vertical translation: 1; one full period is from $t = 0$ to $t = 2 π$

6.1-21 §6.1 source ↗

For the following exercises, graph one full period of each function, starting at $x = 0.$ For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

$f (t) =$ $-$ $\sin (\frac{1}{2} t + \frac{5 π}{3})$

Answer

amplitude: 1; period: $4 π;$ midline: $y = 0;$ maximum: $y = 1$ occurs at $t = 11.52;$ minimum: $y = - 1$ occurs at $t = 5.24;$ phase shift: $- \frac{10 π}{3};$ vertical shift: 0

6.1-23 §6.1 source ↗

For the following exercises, graph one full period of each function, starting at $x = 0.$ For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure 26.

A sinusoidal graph with amplitude of 2, range of [-5, -1], period of 4, and midline at y=-3.

Figure 26

Answer

amplitude: 2; midline:

y = - 3;

period: 4; equation:

f (x) =

2 \sin (\frac{π}{2} x) -

3

6.1-25 §6.1 source ↗

For the following exercises, graph one full period of each function, starting at $x = 0.$ For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 28.

A graph with a cosine parent function with an amplitude of 2, period of 5, midline at y=3, and a range of [1,5].

Figure 28

Answer

amplitude: 2; period: 5; midline:

y = 3;

equation:

f (x) =

-

2 \cos (\frac{2 π}{5} x) +

3

6.1-27 §6.1 source ↗

For the following exercises, graph one full period of each function, starting at $x = 0.$ For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 30.

A graph with cosine parent function, range of function is [-4,4], amplitude of 4, period of 2.

Figure 30

Answer

amplitude: 4; period: 2; midline:

y = 0;

equation:

f (x) =

-

4 \cos (π (x - \frac{π}{2}))

6.1-29 §6.1 source ↗

For the following exercises, graph one full period of each function, starting at $x = 0.$ For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for $x > 0.$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 32.

A graph with cosine parent function. Amplitude 2, period 2, midline y=1

Figure 32

Answer

amplitude: 2; period: 2; midline

y = 1;

equation:

f (x) =

2 \cos (π x) +

1

6.1-31 §6.1 source ↗

For the following exercises, let $f (x) = \sin x .$

On $[0, 2 π),$ solve $f (x) = 0.$

Answer

0, π

6.1-33 §6.1 source ↗

For the following exercises, let $f (x) = \sin x .$

Evaluate $f (\frac{π}{2}) .$

Answer

sin (\frac{π}{2}) = 1

6.1-35 §6.1 source ↗

For the following exercises, let $f (x) = \sin x .$

On $[0, 2 π),$ the maximum value(s) of the function occur(s) at what x-value(s)?

Answer

\frac{π}{2}

6.1-37 §6.1 source ↗

For the following exercises, let $f (x) = \sin x .$

Show that $f (- x) =$ $-$ $f (x) .$ This means that $f (x) = \sin x$ is an odd function and possesses symmetry with respect to ________________.

Answer

f (x) = sin x

is symmetric

6.1-39 §6.1 source ↗

For the following exercises, let $f (x) = \cos x .$

On $[0, 2 π),$ solve $f (x) = \frac{1}{2} .$

Answer

\frac{π}{3}, \frac{5 π}{3}

6.1-41 §6.1 source ↗

For the following exercises, let $f (x) = \cos x .$

On $[0, 2 π),$ find the x-values at which the function has a maximum or minimum value.

Answer

Maximum:

1

at

x = 0

; minimum:

-1

at

x = π

6.1-43 §6.1 source ↗

For the following exercises, let $f (x) = \cos x .$

Graph $h (x) = x + \sin x$ on $[0, 2 π] .$ Explain why the graph appears as it does.

Answer

A linear function is added to a periodic sine function. The graph does not have an amplitude because as the linear function increases without bound the combined function $h (x) = x + sin x$ will increase without bound as well. The graph is bounded between the graphs of $y = x + 1$ and $y = x - 1$ because sine oscillates between −1 and 1.

This image displays a graph of the function h(t) versus t. The horizontal axis represents t, ranging from 0 to 2π, with key markers at π/2, π, and 3π/2. The vertical axis represents h(t), ranging from 0 to 6. The curve starts at the origin (0,0) and rises continuously to approximately (2π, 6). The function exhibits an initial period of increasing slope, followed by a region where the slope decreases (around t=π, where h(t) is about 3), indicating a slowing rate of increase, and then the slope increases again, showing an accelerating rate of increase towards the end of the interval.

6.1-45 §6.1 source ↗

For the following exercises, let $f (x) = \cos x .$

Graph $f (x) = x \sin x$ on $[0, 2 π]$ and verbalize how the graph varies from the graph of $f (x) = \sin x .$

Answer

There is no amplitude because the function is not bounded.

The image presents a graph of a periodic function plotted on a coordinate system featuring an inverted y-axis. The x-axis, labeled X, spans from 0 to 2pi, with key markings at pi/2, pi, 3pi/2, and 2pi. The y-axis, labeled f(x), shows its origin (0) at the top, and positive integer values from 1 to 5 increase as one moves downwards. The curve commences at the point (0,0), which represents a local minimum for the function's value. It then descends, indicating an increase in the function's value, reaching a local maximum of 2 around x=pi/2. Following this, the curve ascends, showing a decrease in value, returning to a local minimum of 0 at x=pi. The function then continues to descend, with its value increasing further to a global maximum of 5 around x=3pi/2, before finally ascending back to a local minimum of 0 at x=2pi, thereby completing one full period of oscillation.

6.1-47 §6.1 source ↗

For the following exercises, let $f (x) = \cos x .$

Graph $f (x) = \frac{\sin x}{x}$ on the window $[−5 π, 5 π]$ and explain what the graph shows.

Answer

The graph is symmetric with respect to the y-axis and there is no amplitude because the function’s bounds decrease as $| x |$ grows. There appears to be a horizontal asymptote at $y = 0$ .

A graph showing a damped oscillatory function resembling sin(x)/x. The x-axis spans from -5π to 5π, and the y-axis from -2 to 2. The curve peaks at (0,1) and crosses the x-axis at integer multiples of π.

Try It #1 §6.2 source ↗

Sketch a graph of

f (x) =

3 \tan (\frac{π}{6} x) .

Answer

A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.

Try It #2 §6.2 source ↗

How would the graph in Example 2 look different if we made

A = 2

instead of

−2 ?

Answer

It would be reflected across the line

y = - 1,

becoming an increasing function.

Try It #3 §6.2 source ↗

Find a formula for the function in Figure 5.

A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi/4, -pi/4, pi/4, and 3pi/4.

Figure 5

Answer

g (x) =

4 \tan (2 x)

Try It #4 §6.2 source ↗

Graph one period of

f (x) =

-

2.5 \sec (0.4 x) .

Answer

This is a vertical reflection of the preceding graph because $A$ is negative.

A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.

Try It #5 §6.2 source ↗

Graph one period of

f (x) =

-

6 \sec (4 x + 2) -

8.

Answer

A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi/20 and one approximately at 3pi/16.

Try It #6 §6.2 source ↗

Graph one period of

f (x) =

0.5 \csc (2 x) .

Answer

Try It #7 §6.2 source ↗

Given the graph of $f (x) =$ $2 \cos (\frac{π}{2} x) +$ $1$ shown in Figure 12, sketch the graph of $g (x) =$ $2 \sec (\frac{π}{2} x) +$ $1$ on the same axes.

A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.

Figure 12

Answer

A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.

6.2-1 §6.2 source ↗

Explain how the graph of the sine function can be used to graph

y = \csc x .

Answer

Since

y = \csc x

is the reciprocal function of

y = \sin x,

you can plot the reciprocal of the coordinates on the graph of

y = \sin x

to obtain the y-coordinates of

y = \csc x .

The x-intercepts of the graph

y = \sin x

are the vertical asymptotes for the graph of

y = \csc x .

6.2-3 §6.2 source ↗

Explain why the period of

\tan x

is equal to

π .

Answer

Answers will vary. Using the unit circle, one can show that

\tan (x + π) =

\tan x .

6.2-5 §6.2 source ↗

How does the period of

y = \csc x

compare with the period of

y = \sin x ?

Answer

The period is the same:

2 π .

6.2-7 §6.2 source ↗

For the following exercises, match each trigonometric function with one of the following graphs.

The image presents four distinct graphs, each labeled with a Roman numeral and representing a trigonometric function. Graph I shows the tangent function, y = tan(x), characterized by increasing curves passing through the origin with vertical asymptotes at x = ±π/2. Graph II depicts the cosecant function, y = csc(x), with U-shaped branches opening upwards and downwards, and vertical asymptotes at integer multiples of π, such as x = π. Graph III illustrates the cotangent function, y = cot(x), featuring decreasing curves that cross the x-axis at x = π/2, with vertical asymptotes at integer multiples of π. Graph IV represents the secant function, y = sec(x), also showing U-shaped branches opening upwards and downwards, with vertical asymptotes at odd multiples of π/2, such as x = π/2 and x = 3π/2.

Figure 18

$f (x) = \sec x$

Answer

IV

6.2-9 §6.2 source ↗

For the following exercises, match each trigonometric function with one of the following graphs.

Figure 18

$f (x) = \cot x$

Answer

III

6.2-11 §6.2 source ↗

For the following exercises, find the period and horizontal shift of each of the functions.

$h (x) =$ $2 \sec (\frac{π}{4} (x + 1))$

Answer

period: 8; horizontal shift: 1 unit to left

6.2-13 §6.2 source ↗

For the following exercises, evaluate the transformed functions.

If $\tan x = - 1.5,$ find $\tan (- x) .$

Answer

1.5

6.2-15 §6.2 source ↗

For the following exercises, evaluate the transformed functions.

If $\csc x = - 5,$ find $\csc (- x) .$

Answer

5

6.2-17 §6.2 source ↗

For the following exercises, rewrite each expression such that the argument $x$ is positive.

$\cot (- x) \cos (- x) +$ $\sin (- x)$

Answer

-

\cot x \cos x -

\sin x

6.2-19 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$f (x) =$ $2 \tan (4 x - 32)$

Answer

A graph of two periods of a modified tangent function. There are two vertical asymptotes.

stretching factor: 2; period: $\frac{π}{4};$ asymptotes: $x =$ $\frac{1}{4} (\frac{π}{2} + π k) +$ $8,$ $where k is an integer$

6.2-21 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$m (x) =$ $6 \csc (\frac{π}{3} x + π)$

Answer

A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6.

stretching factor: 6; period: 6; asymptotes: $x =$ $3 k,$ $where k is an integer$

6.2-23 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$p (x) =$ $\tan (x - \frac{π}{2})$

Answer

A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi.

stretching factor: 1; period: $π;$ asymptotes: $x =$ $π k,$ $where k is an integer$

6.2-25 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$f (x) =$ $\tan (x + \frac{π}{4})$

Answer

A graph of two periods of a modified tangent function. Three vertical asymptiotes shown.

Stretching factor: 1; period: $π;$ asymptotes: $x =$ $\frac{π}{4} +$ $π k,$ $where k is an integer$

6.2-27 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$f (x) =$ $2 \csc (x)$

Answer

A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi.

stretching factor: 2; period: $2 π;$ asymptotes: $x =$ $π k,$ $where k is an integer$

6.2-29 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$f (x) =$ $4 \sec (3 x)$

Answer

A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi/2, -pi/6, pi/6, and pi/2.

stretching factor: 4; period: $\frac{2 π}{3};$ asymptotes: $x =$ $\frac{π}{6} k,$ $where k is an odd integer$

6.2-31 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$f (x) =$ $7 \sec (5 x)$

Answer

A graph of two periods of a modified secant function. There are four vertical asymptotes all pi/5 apart.

stretching factor: 7; period: $\frac{2 π}{5};$ asymptotes: $x =$ $\frac{π}{10} k,$ $where k is an odd integer$

6.2-33 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$f (x) =$ $2 \csc (x + \frac{π}{4}) -$ $1$

Answer

A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.

stretching factor: 2; period: $2 π;$ asymptotes: $x =$ $-$ $\frac{π}{4} +$ $π k,$ $where k is an integer$

6.2-35 §6.2 source ↗

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

$f (x) =$ $\frac{7}{5} \csc (x - \frac{π}{4})$

Answer

A graph of a modified cosecant function. Four vertical asymptotes.

stretching factor: $\frac{7}{5};$ period: $2 π;$ asymptotes: $x =$ $\frac{π}{4} +$ $π k,$ $where k is an integer$

6.2-37 §6.2 source ↗

For the following exercises, find and graph two periods of the periodic function with the given stretching factor, $| A |,$ period, and phase shift.

A tangent curve, $A = 1,$ period of $\frac{π}{3};$ and phase shift $(h, k) =$ $(\frac{π}{4}, 2)$

Answer

$y =$ $\tan (3 (x - \frac{π}{4})) +$ $2$

A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi/4 and pi/12.

6.2-39 §6.2 source ↗

For the following exercises, find an equation for the graph of each function.

A graph of two periods of a modified cosecant function, with asymptotes at multiples of pi/2.

Answer

f (x) =

\csc (2 x)

6.2-41 §6.2 source ↗

For the following exercises, find an equation for the graph of each function.

A graph of a modified cosecant function. Vertical asymptotes at multiples of pi/4.

Answer

f (x) =

\csc (4 x)

6.2-43 §6.2 source ↗

For the following exercises, find an equation for the graph of each function.

A graph of a modified cosecant function. Vertical asymptotyes at multiples of pi.

Answer

f (x) = 2 \csc x

6.2-45 §6.2 source ↗

For the following exercises, find an equation for the graph of each function.

graph of two periods of a modified tangent function. Vertical asymptotes at x=-0.005 and x=0.005.

Answer

f (x) =

\frac{1}{2} \tan (100 π x)

6.2-47 §6.2 source ↗

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc x$ as $\frac{1}{\sin x} .$

$f (x) =$ $| \cot (x) |$

Answer

A graph of the absolute value of the cotangent function. Range is 0 to infinity.

6.2-49 §6.2 source ↗

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc x$ as $\frac{1}{\sin x} .$

$f (x) =$ $\frac{\csc (x)}{\sec (x)}$

Answer

6.2-51 §6.2 source ↗

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc x$ as $\frac{1}{\sin x} .$

$f (x) =$ $\sec (0.001 x)$

Answer

A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi.

6.2-53 §6.2 source ↗

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc x$ as $\frac{1}{\sin x} .$

$f (x) =$ $\sin^{2} x +$ $\cos^{2} x$

Answer

A graph showing a horizontal blue line at y=1 on a Cartesian coordinate system. The x-axis is labeled with multiples of pi/2, and the y-axis is labeled with integers from -2 to 2.

6.2-55 §6.2 source ↗

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc x$ as $\frac{1}{\sin x} .$

Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let $x,$ measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and $x$ is measured negative to the left and positive to the right. (See Figure 19.) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance $d (x),$ in kilometers, from the fisherman to the boat is given by the function $d (x) =$ $1.5 \sec (x) .$

ⓐWhat is a reasonable domain for $d (x) ?$

ⓑGraph $d (x)$ on this domain.

ⓒFind and discuss the meaning of any vertical asymptotes on the graph of $d (x) .$

ⓓCalculate and interpret $d (- \frac{π}{3}) .$ Round to the second decimal place.

ⓔCalculate and interpret $d (\frac{π}{6}) .$ Round to the second decimal place.

ⓕWhat is the minimum distance between the fisherman and the boat? When does this occur?

An illustration of a man and the distance he is away from a boat.

Figure 19

Answer

ⓐ $(- \frac{π}{2}, \frac{π}{2});$

ⓑ A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2.

ⓒ $x = - \frac{π}{2}$ and $x = \frac{π}{2};$ the distance grows without bound as $| x |$ approaches $\frac{π}{2}$ —i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;

ⓓ3; when $x = - \frac{π}{3},$ the boat is 3 km away;

ⓔ 1.73; when $x = \frac{π}{6},$ the boat is about 1.73 km away;

ⓕ 1.5 km; when $x = 0$

6.2-57 §6.2 source ↗

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc x$ as $\frac{1}{\sin x} .$

A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after $x$ seconds is $\frac{π}{120} x .$

ⓐWrite a function expressing the altitude $h (x),$ in miles, of the rocket above the ground after $x$ seconds. Ignore the curvature of the Earth.

ⓑGraph $h (x)$ on the interval $(0, 60) .$

ⓒEvaluate and interpret the values $h (0)$ and $h (30) .$

ⓓWhat happens to the values of $h (x)$ as $x$ approaches 60 seconds? Interpret the meaning of this in terms of the problem.

Answer

ⓐ $h (x) =$ $2 \tan (\frac{π}{120} x);$

ⓑ An exponentially increasing function with a vertical asymptote at x=60.

ⓓAs $x$ approaches 60 seconds, the values of $h (x)$ grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.

Try It #1 §6.3 source ↗

Given

\cos (0.5) \approx

0.8776,

write a relation involving the inverse cosine.

Answer

\arccos (0.8776) \approx

0.5

Try It #2 §6.3 source ↗

Evaluate each of the following.

ⓐ ${sin}^{−1} (−1)$

ⓑ $\tan^{−1} (−1)$

ⓓ $\cos^{−1} (\frac{1}{2})$

Answer

ⓐ $- \frac{π}{2};$

ⓑ $- \frac{π}{4};$

ⓓ $\frac{π}{3}$

Try It #3 §6.3 source ↗

Evaluate

\cos^{- 1} (- 0.4)

using a calculator.

Answer

1.9823 or 113.578°

Try It #4 §6.3 source ↗

Solve the triangle in Figure 9 for the angle $θ .$

An illustration of a right triangle with the angle theta. Opposite to the angle theta is a side with a length of 6 and a hypoteneuse of length 10.

Figure 9

Answer

\sin^{−1} (0.6) =

36.87° =

0.6435

radians

Try It #5 §6.3 source ↗

Evaluate

\tan^{- 1} (\tan (\frac{π}{8})) and \tan^{- 1} (\tan (\frac{11 π}{9})) .

Answer

\frac{π}{8}; \frac{2 π}{9}

Try It #6 §6.3 source ↗

Evaluate

\cos^{- 1} (\sin (- \frac{11 π}{4})) .

Answer

\frac{3 π}{4}

Try It #7 §6.3 source ↗

Evaluate

\cos (\tan^{- 1} (\frac{5}{12})) .

Answer

\frac{12}{13}

Try It #8 §6.3 source ↗

Evaluate

\cos (\sin^{- 1} (\frac{7}{9})) .

Answer

\frac{4 \sqrt{2}}{9}

Try It #9 §6.3 source ↗

Find a simplified expression for

\sin (\tan^{- 1} (4 x))

for

- \frac{1}{4} \leq x \leq \frac{1}{4} .

Answer

\frac{4 x}{\sqrt{16 x^{2} + 1}}

6.3-1 §6.3 source ↗

Why do the functions

f (x) = \sin^{- 1} x

and

g (x) = \cos^{- 1} x

have different ranges?

Answer

The function

y = \sin x

is one-to-one on

[- \frac{π}{2}, \frac{π}{2}];

thus, this interval is the range of the inverse function of

y = \sin x,

f (x) =

\sin^{- 1} x .

The function

y = \cos x

is one-to-one on

[0, π];

thus, this interval is the range of the inverse function of

y =

\cos x,

f (x) =

\cos^{- 1} x .

6.3-3 §6.3 source ↗

Explain the meaning of

\frac{π}{6} =

\arcsin (0.5) .

Answer

\frac{π}{6}

is the radian measure of an angle between

- \frac{π}{2}

and

\frac{π}{2}

whose sine is 0.5.

6.3-5 §6.3 source ↗

Why must the domain of the sine function,

\sin x,

be restricted to

[- \frac{π}{2}, \frac{π}{2}]

for the inverse sine function to exist?

Answer

In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval

[- \frac{π}{2}, \frac{π}{2}]

so that it is one-to-one and possesses an inverse.

6.3-7 §6.3 source ↗

Determine whether the following statement is true or false and explain your answer:

\arccos (- x) =

π -

\arccos x .

Answer

True . The angle,

θ_{1}

that equals

\arccos (- x)

,

x > 0

, will be a second quadrant angle with reference angle,

θ_{2}

, where

θ_{2}

equals

\arccos x

,

x > 0

. Since

θ_{2}

is the reference angle for

θ_{1}

,

θ_{2} = π - θ_{1}

and

\arccos (- x)

=

π - \arccos x

-

6.3-9 §6.3 source ↗

For the following exercises, evaluate the expressions.

$\sin^{- 1} (- \frac{1}{2})$

Answer

- \frac{π}{6}

6.3-11 §6.3 source ↗

For the following exercises, evaluate the expressions.

$\cos^{- 1} (- \frac{\sqrt{2}}{2})$

Answer

\frac{3 π}{4}

6.3-13 §6.3 source ↗

For the following exercises, evaluate the expressions.

$\tan^{- 1} (- \sqrt{3})$

Answer

- \frac{π}{3}

6.3-15 §6.3 source ↗

For the following exercises, evaluate the expressions.

$\tan^{- 1} (\sqrt{3})$

Answer

\frac{π}{3}

6.3-17 §6.3 source ↗

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

$\cos^{- 1} (- 0.4)$

Answer

1.98

6.3-19 §6.3 source ↗

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

$\arccos (\frac{3}{5})$

Answer

0.93

6.3-21 §6.3 source ↗

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.

$\tan^{- 1} (6)$

Answer

1.41

6.3-23 §6.3 source ↗

For the following exercises, find the angle $θ$ in the given right triangle. Round answers to the nearest hundredth.

An illustration of a right triangle with angle theta. Adjacent the angle theta is a side of length 19. Opposite the angle theta is a side with length 12.

Answer

0.56 radians

6.3-25 §6.3 source ↗

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} (\sin (π))$

Answer

0

6.3-27 §6.3 source ↗

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} (\sin (\frac{π}{3}))$

Answer

0.71

6.3-29 §6.3 source ↗

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} (\sin (\frac{4 π}{3}))$

Answer

-0.71

6.3-31 §6.3 source ↗

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} (\sin (\frac{- 5 π}{2}))$

Answer

- \frac{π}{4}

6.3-33 §6.3 source ↗

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\sin (\cos^{- 1} (\frac{3}{5}))$

Answer

0.8

6.3-35 §6.3 source ↗

For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\cos (\tan^{- 1} (\frac{12}{5}))$

Answer

\frac{5}{13}

6.3-37 §6.3 source ↗

For the following exercises, find the exact value of the expression in terms of $x$ with the help of a reference triangle.

$\tan (\sin^{- 1} (x - 1))$

Answer

\frac{x - 1}{\sqrt{- x^{2} + 2 x}}

6.3-39 §6.3 source ↗

For the following exercises, find the exact value of the expression in terms of $x$ with the help of a reference triangle.

$\cos (\sin^{- 1} (\frac{1}{x}))$

Answer

\frac{\sqrt{x^{2} - 1}}{x}

6.3-41 §6.3 source ↗

For the following exercises, find the exact value of the expression in terms of $x$ with the help of a reference triangle.

$\tan (\sin^{- 1} (x + \frac{1}{2}))$

Answer

\frac{x + 0.5}{\sqrt{- x^{2} - x + \frac{3}{4}}}

6.3-43 §6.3 source ↗