# 6.3 Inverse trigonometric functions  (Page 9/15)

 Page 9 / 15

What are the amplitude, period, and phase shift for the function?

amplitude: 8,000; period: 10; phase shift: 0

Over this domain, when does the population reach 18,000? 13,000?

What is the predicted population in 2007? 2010?

In 2007, the predicted population is 4,413. In 2010, the population will be 11,924.

For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

Suppose the graph of the displacement function is shown in [link] , where the values on the x -axis represent the time in seconds and the y -axis represents the displacement in inches. Give the equation that models the vertical displacement of the weight on the spring.

At time = 0, what is the displacement of the weight?

5 in.

At what time does the displacement from the equilibrium point equal zero?

What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?

10 seconds

## Inverse Trigonometric Functions

For the following exercises, find the exact value without the aid of a calculator.

${\mathrm{sin}}^{-1}\left(1\right)$

${\mathrm{cos}}^{-1}\left(\frac{\sqrt{3}}{2}\right)$

$\frac{\pi }{6}$

${\mathrm{tan}}^{-1}\left(-1\right)$

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

$\frac{\pi }{4}$

${\mathrm{sin}}^{-1}\left(\frac{-\sqrt{3}}{2}\right)$

${\mathrm{sin}}^{-1}\left(\mathrm{cos}\left(\frac{\pi }{6}\right)\right)$

$\frac{\pi }{3}$

${\mathrm{cos}}^{-1}\left(\mathrm{tan}\left(\frac{3\pi }{4}\right)\right)$

$\mathrm{sin}\left({\mathrm{sec}}^{-1}\left(\frac{3}{5}\right)\right)$

No solution

$\mathrm{cot}\left({\mathrm{sin}}^{-1}\left(\frac{3}{5}\right)\right)$

$\mathrm{tan}\left({\mathrm{cos}}^{-1}\left(\frac{5}{13}\right)\right)$

$\frac{12}{5}$

$\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(\frac{x}{x+1}\right)\right)$

Graph $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sec}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right)\text{\hspace{0.17em}}$ and explain any observations.

The graphs are not symmetrical with respect to the line $\text{\hspace{0.17em}}y=x.\text{\hspace{0.17em}}$ They are symmetrical with respect to the $\text{\hspace{0.17em}}y$ -axis.

Graph $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{csc}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and explain any observations.

Graph the function $f\text{\hspace{0.17em}}\left(x\right)=\frac{x}{1}-\frac{{x}^{3}}{3!}+\frac{{x}^{5}}{5!}-\frac{{x}^{7}}{7!}\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left[-1,1\right]\text{\hspace{0.17em}}$ and compare the graph to the graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on the same interval. Describe any observations.

The graphs appear to be identical.

## Chapter practice test

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

$f\left(x\right)=0.5\mathrm{sin}\text{\hspace{0.17em}}x$

amplitude: 0.5; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$

$f\left(x\right)=5\mathrm{cos}\text{\hspace{0.17em}}x$

$f\left(x\right)=5\mathrm{sin}\text{\hspace{0.17em}}x$

amplitude: 5; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0$

$f\left(x\right)=\mathrm{sin}\left(3x\right)$

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

amplitude: 1; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=1$

$f\left(x\right)=5\mathrm{sin}\left(3\left(x-\frac{\pi }{6}\right)\right)+4$

$f\left(x\right)=3\mathrm{cos}\left(\frac{1}{3}x-\frac{5\pi }{6}\right)$

amplitude: 3; period: $\text{\hspace{0.17em}}6\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0$

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

$f\left(x\right)=-2\mathrm{tan}\left(x-\frac{7\pi }{6}\right)+2$

amplitude: none; period: midline: asymptotes: where is an integer

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

$f\left(x\right)=5\mathrm{csc}\left(3x\right)$

amplitude: none; period: midline: asymptotes: where is an integer

$f\left(x\right)=\pi \mathrm{sec}\left(\frac{\pi }{2}x\right)$

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

amplitude: none; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=-3$

For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

Give in terms of a sine function.

Give in terms of a sine function.

amplitude: 2; period: 2; midline: $\text{\hspace{0.17em}}y=0;$ $f\left(x\right)=2\mathrm{sin}\left(\pi \left(x-1\right)\right)$

Give in terms of a tangent function.

For the following exercises, find the amplitude, period, phase shift, and midline.

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

amplitude: 1; period: 12; phase shift: $\text{\hspace{0.17em}}-6;\text{\hspace{0.17em}}$ midline $\text{\hspace{0.17em}}y=-3$

$y=8\mathrm{sin}\left(\frac{7\pi }{6}x+\frac{7\pi }{2}\right)+6$

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the number of hours since midnight, find a function for the temperature, $\text{\hspace{0.17em}}D,\text{\hspace{0.17em}}$ in terms of $\text{\hspace{0.17em}}t.$

$D\left(t\right)=68-12\mathrm{sin}\left(\frac{\pi }{12}x\right)$

Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

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

$g\left(x\right)=3\mathrm{tan}\left(6x+42\right)$

period: $\text{\hspace{0.17em}}\frac{\pi }{6};\text{\hspace{0.17em}}$ horizontal shift: $\text{\hspace{0.17em}}-7$

$n\left(x\right)=4\mathrm{csc}\left(\frac{5\pi }{3}x-\frac{20\pi }{3}\right)$

Write the equation for the graph in [link] in terms of the secant function and give the period and phase shift.

$f\left(x\right)=\mathrm{sec}\left(\pi x\right);\text{\hspace{0.17em}}$ period: 2; phase shift: 0

If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\mathrm{tan}\left(-x\right).$

If $\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=4,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\mathrm{sec}\left(-x\right).$

$4$

For the following exercises, graph the functions on the specified window and answer the questions.

Graph $\text{\hspace{0.17em}}m\left(x\right)=\mathrm{sin}\left(2x\right)+\mathrm{cos}\left(3x\right)\text{\hspace{0.17em}}$ on the viewing window $\text{\hspace{0.17em}}\left[-10,10\right]\text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}\left[-3,3\right].\text{\hspace{0.17em}}$ Approximate the graph’s period.

Graph $\text{\hspace{0.17em}}n\left(x\right)=0.02\mathrm{sin}\left(50\pi x\right)\text{\hspace{0.17em}}$ on the following domains in $\text{\hspace{0.17em}}x:$ $\left[0,1\right]\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left[0,3\right].\text{\hspace{0.17em}}$ Suppose this function models sound waves. Why would these views look so different?

The views are different because the period of the wave is $\text{\hspace{0.17em}}\frac{1}{25}.\text{\hspace{0.17em}}$ Over a bigger domain, there will be more cycles of the graph.

Graph $\text{\hspace{0.17em}}f\left(x\right)=\frac{\mathrm{sin}\text{\hspace{0.17em}}x}{x}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-0.5,0.5\right]\text{\hspace{0.17em}}$ and explain any observations.

For the following exercises, let $\text{\hspace{0.17em}}f\left(x\right)=\frac{3}{5}\mathrm{cos}\left(6x\right).$

What is the largest possible value for $\text{\hspace{0.17em}}f\left(x\right)?$

$\frac{3}{5}$

What is the smallest possible value for $\text{\hspace{0.17em}}f\left(x\right)?$

Where is the function increasing on the interval $\text{\hspace{0.17em}}\left[0,2\pi \right]?$

On the approximate intervals $\text{\hspace{0.17em}}\left(0.5,1\right),\left(1.6,2.1\right),\left(2.6,3.1\right),\left(3.7,4.2\right),\left(4.7,5.2\right),\left(5.6,6.28\right)$

For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

Sine curve with amplitude 3, period $\text{\hspace{0.17em}}\frac{\pi }{3},\text{\hspace{0.17em}}$ and phase shift $\text{\hspace{0.17em}}\left(h,k\right)=\left(\frac{\pi }{4},2\right)$

Cosine curve with amplitude 2, period $\text{\hspace{0.17em}}\frac{\pi }{6},\text{\hspace{0.17em}}$ and phase shift $\text{\hspace{0.17em}}\left(h,k\right)=\left(-\frac{\pi }{4},3\right)$

$f\left(x\right)=2\mathrm{cos}\left(12\left(x+\frac{\pi }{4}\right)\right)+3$

For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

$f\left(x\right)=5\mathrm{cos}\left(3x\right)+4\mathrm{sin}\left(2x\right)$

$f\left(x\right)={e}^{\mathrm{sin}t}$

This graph is periodic with a period of $\text{\hspace{0.17em}}2\pi .$

For the following exercises, find the exact value.

${\mathrm{sin}}^{-1}\left(\frac{\sqrt{3}}{2}\right)$

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

$\frac{\pi }{3}$

${\mathrm{cos}}^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

${\mathrm{cos}}^{-1}\left(\mathrm{sin}\left(\pi \right)\right)$

$\frac{\pi }{2}$

${\mathrm{cos}}^{-1}\left(\mathrm{tan}\left(\frac{7\pi }{4}\right)\right)$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(1-2x\right)\right)$

$\sqrt{1-{\left(1-2x\right)}^{2}}$

${\mathrm{cos}}^{-1}\left(-0.4\right)$

$\mathrm{cos}\left({\mathrm{tan}}^{-1}\left({x}^{2}\right)\right)$

$\frac{1}{\sqrt{1+{x}^{4}}}$

For the following exercises, suppose $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=\frac{x}{x+1}.$

$\mathrm{tan}\text{\hspace{0.17em}}t$

$\mathrm{csc}\text{\hspace{0.17em}}t$

$\frac{x+1}{x}$

Given [link] , find the measure of angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to three decimal places. Answer in radians.

For the following exercises, determine whether the equation is true or false.

$\mathrm{arcsin}\left(\mathrm{sin}\left(\frac{5\pi }{6}\right)\right)=\frac{5\pi }{6}$

False

$\mathrm{arccos}\left(\mathrm{cos}\left(\frac{5\pi }{6}\right)\right)=\frac{5\pi }{6}$

The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.

#### Questions & Answers

A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?