# 5.6 Inverse trigonometric functions  (Page 8/15)

 Page 8 / 15

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 $\text{\hspace{0.17em}}\left[0,40\right]$ .

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!