# 5.6 Inverse trigonometric functions  (Page 7/15)

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Discuss why this statement is incorrect: $\text{\hspace{0.17em}}\mathrm{arccos}\left(\mathrm{cos}\text{\hspace{0.17em}}x\right)=x\text{\hspace{0.17em}}$ for all $\text{\hspace{0.17em}}x.$

Determine whether the following statement is true or false and explain your answer: $\mathrm{arccos}\left(-x\right)=\pi -\mathrm{arccos}\text{\hspace{0.17em}}x.$

True . The angle, $\text{\hspace{0.17em}}{\theta }_{1}\text{\hspace{0.17em}}$ that equals $\text{\hspace{0.17em}}\mathrm{arccos}\left(-x\right)\text{\hspace{0.17em}}$ , $\text{\hspace{0.17em}}x>0\text{\hspace{0.17em}}$ , will be a second quadrant angle with reference angle, $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ , where $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ equals $\text{\hspace{0.17em}}\mathrm{arccos}x$ , $x>0\text{\hspace{0.17em}}$ . Since $\text{\hspace{0.17em}}{\theta }_{2}\text{\hspace{0.17em}}$ is the reference angle for $\text{\hspace{0.17em}}{\theta }_{1}$ , ${\theta }_{2}=\pi -{\theta }_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{arccos}\left(-x\right)\text{\hspace{0.17em}}$ = $\text{\hspace{0.17em}}\pi -\mathrm{arccos}x$ -

## Algebraic

For the following exercises, evaluate the expressions.

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

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

$-\frac{\pi }{6}$

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

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

$\frac{3\pi }{4}$

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

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

$-\frac{\pi }{3}$

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

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

$\frac{\pi }{3}$

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

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

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

1.98

$\mathrm{arcsin}\left(0.23\right)$

$\mathrm{arccos}\left(\frac{3}{5}\right)$

0.93

${\mathrm{cos}}^{-1}\left(0.8\right)$

${\mathrm{tan}}^{-1}\left(6\right)$

1.41

For the following exercises, find the angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in the given right triangle. Round answers to the nearest hundredth.

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

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

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

0

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

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

0.71

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

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

-0.71

${\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\frac{5\pi }{6}\right)\right)$

${\mathrm{tan}}^{-1}\left(\mathrm{sin}\left(\frac{-5\pi }{2}\right)\right)$

$-\frac{\pi }{4}$

$\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(\frac{4}{5}\right)\right)$

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

0.8

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

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

$\frac{5}{13}$

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

For the following exercises, find the exact value of the expression in terms of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with the help of a reference triangle.

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

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

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

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

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

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

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

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

## Extensions

For the following exercises, evaluate the expression without using a calculator. Give the exact value.

$\frac{{\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{cos}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{sin}}^{-1}\left(\frac{\sqrt{3}}{2}\right)-{\mathrm{cos}}^{-1}\left(1\right)}{{\mathrm{cos}}^{-1}\left(\frac{\sqrt{3}}{2}\right)-{\mathrm{sin}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)-{\mathrm{sin}}^{-1}\left(0\right)}$

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

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

$\frac{\sqrt{2x+1}}{x+1}$

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

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

$\frac{\sqrt{2x+1}}{x}$

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

${\mathrm{tan}}^{-1}\left(\frac{x}{\sqrt{2x+1}}\right)$

$t$

## Graphical

Graph $\text{\hspace{0.17em}}y={\mathrm{sin}}^{-1}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

Graph $\text{\hspace{0.17em}}y=\mathrm{arccos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

domain $\text{\hspace{0.17em}}\left[-1,1\right];\text{\hspace{0.17em}}$ range $\text{\hspace{0.17em}}\left[0,\pi \right]\text{\hspace{0.17em}}$

Graph one cycle of $\text{\hspace{0.17em}}y={\mathrm{tan}}^{-1}x\text{\hspace{0.17em}}$ and state the domain and range of the function.

For what value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x={\mathrm{sin}}^{-1}x?\text{\hspace{0.17em}}$ Use a graphing calculator to approximate the answer.

approximately $\text{\hspace{0.17em}}x=0.00\text{\hspace{0.17em}}$

For what value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x={\mathrm{cos}}^{-1}x?\text{\hspace{0.17em}}$ Use a graphing calculator to approximate the answer.

## Real-world applications

Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-ﬂoor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?

Suppose you drive 0.6 miles on a road so that the vertical distance changes from 0 to 150 feet. What is the angle of elevation of the road?

An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

Without using a calculator, approximate the value of $\text{\hspace{0.17em}}\mathrm{arctan}\left(10,000\right).\text{\hspace{0.17em}}$ Explain why your answer is reasonable.

A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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