# 5.6 Inverse trigonometric functions  (Page 8/15)

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The line $\text{\hspace{0.17em}}y=\frac{3}{5}x\text{\hspace{0.17em}}$ passes through the origin in the x , y -plane. What is the measure of the angle that the line makes with the positive x -axis?

The line $\text{\hspace{0.17em}}y=\frac{-3}{7}x\text{\hspace{0.17em}}$ passes through the origin in the x , y -plane. What is the measure of the angle that the line makes with the negative x -axis?

What percentage grade should a road have if the angle of elevation of the road is 4 degrees? (The percentage grade is defined as the change in the altitude of the road over a 100-foot horizontal distance. For example a 5% grade means that the road rises 5 feet for every 100 feet of horizontal distance.)

A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder's angle of elevation to be between 35 and 45 degrees, does the placement of this ladder satisfy safety specifications?

No. The angle the ladder makes with the horizontal is 60 degrees.

Suppose a 15-foot ladder leans against the side of a house so that the angle of elevation of the ladder is 42 degrees. How far is the foot of the ladder from the side of the house?

## Graphs of the Sine and Cosine Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

$f\left(x\right)=-3\mathrm{cos}\text{\hspace{0.17em}}x+3$

amplitude: 3; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=3;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=\frac{1}{4}\mathrm{sin}\text{\hspace{0.17em}}x$

$f\left(x\right)=3\mathrm{cos}\left(x+\frac{\pi }{6}\right)$

amplitude: 3; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=-2\mathrm{sin}\left(x-\frac{2\pi }{3}\right)$

$f\left(x\right)=3\mathrm{sin}\left(x-\frac{\pi }{4}\right)-4$

amplitude: 3; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=-4;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=2\left(\mathrm{cos}\left(x-\frac{4\pi }{3}\right)+1\right)$

$f\left(x\right)=6\mathrm{sin}\left(3x-\frac{\pi }{6}\right)-1$

amplitude: 6; period: $\text{\hspace{0.17em}}\frac{2\pi }{3};\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=-1;\text{\hspace{0.17em}}$ no asymptotes

$f\left(x\right)=-100\mathrm{sin}\left(50x-20\right)$

## Graphs of the Other Trigonometric Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

$f\left(x\right)=\mathrm{tan}\text{\hspace{0.17em}}x-4$

stretching factor: none; period: midline: asymptotes: where is an integer

$f\left(x\right)=2\mathrm{tan}\left(x-\frac{\pi }{6}\right)$

$f\left(x\right)=-3\mathrm{tan}\left(4x\right)-2$

stretching factor: 3; period: midline: asymptotes: $x=\frac{\pi }{8}+\frac{\pi }{4}k,$ where is an integer

$f\left(x\right)=0.2\mathrm{cos}\left(0.1x\right)+0.3$

For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.

$f\left(x\right)=\frac{1}{3}\mathrm{sec}\text{\hspace{0.17em}}x$

amplitude: none; period: $2\pi ;$ no phase shift; asymptotes: where is an odd integer

$f\left(x\right)=3\mathrm{cot}\text{\hspace{0.17em}}x$

$f\left(x\right)=4\mathrm{csc}\left(5x\right)$

amplitude: none; period: no phase shift; asymptotes: where is an integer

$f\left(x\right)=8\mathrm{sec}\left(\frac{1}{4}x\right)$

$f\left(x\right)=\frac{2}{3}\mathrm{csc}\left(\frac{1}{2}x\right)$

amplitude: none; period: no phase shift; asymptotes: where is an integer

$f\left(x\right)=-\mathrm{csc}\left(2x+\pi \right)$

For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $\text{\hspace{0.17em}}y=12,000+8,000\mathrm{sin}\left(0.628x\right),\text{\hspace{0.17em}}$ where the domain is the years since 1980 and the range is the population of the city.

What is the largest and smallest population the city may have?

largest: 20,000; smallest: 4,000

Graph the function on the domain of $\text{\hspace{0.17em}}\left[0,40\right]$ .

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