# 5.6 Inverse trigonometric functions  (Page 6/15)

 Page 6 / 15

Find a simplified expression for $\text{\hspace{0.17em}}\mathrm{sin}\left({\mathrm{tan}}^{-1}\left(4x\right)\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}-\frac{1}{4}\le x\le \frac{1}{4}.$

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

Access this online resource for additional instruction and practice with inverse trigonometric functions.

Visit this website for additional practice questions from Learningpod.

## Key concepts

• An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
• Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
• For any trigonometric function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x={f}^{-1}\left(y\right),\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}f\left(x\right)=y.\text{\hspace{0.17em}}$ However, $\text{\hspace{0.17em}}f\left(x\right)=y\text{\hspace{0.17em}}$ only implies $\text{\hspace{0.17em}}x={f}^{-1}\left(y\right)\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in the restricted domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ See [link] .
• Special angles are the outputs of inverse trigonometric functions for special input values; for example, $\text{\hspace{0.17em}}\frac{\pi }{4}={\mathrm{tan}}^{-1}\left(1\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{\pi }{6}={\mathrm{sin}}^{-1}\left(\frac{1}{2}\right).$ See [link] .
• A calculator will return an angle within the restricted domain of the original trigonometric function. See [link] .
• Inverse functions allow us to find an angle when given two sides of a right triangle. See [link] .
• In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, $\text{\hspace{0.17em}}\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(x\right)\right)=\sqrt{1-{x}^{2}}.\text{\hspace{0.17em}}$ See [link] .
• If the inside function is a trigonometric function, then the only possible combinations are $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(\mathrm{cos}\text{\hspace{0.17em}}x\right)=\frac{\pi }{2}-x\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}0\le x\le \pi \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(\mathrm{sin}\text{\hspace{0.17em}}x\right)=\frac{\pi }{2}-x\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}-\frac{\pi }{2}\le x\le \frac{\pi }{2}.$ See [link] and [link] .
• When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See [link] .
• When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See [link] .

## Verbal

Why do the functions $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{sin}}^{-1}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{cos}}^{-1}x\text{\hspace{0.17em}}$ have different ranges?

The function $\text{\hspace{0.17em}}y=\mathrm{sin}x\text{\hspace{0.17em}}$ is one-to-one on $\text{\hspace{0.17em}}\left[-\frac{\pi }{2},\frac{\pi }{2}\right];\text{\hspace{0.17em}}$ thus, this interval is the range of the inverse function of $\text{\hspace{0.17em}}y=\mathrm{sin}x,$ $f\left(x\right)={\mathrm{sin}}^{-1}x.\text{\hspace{0.17em}}$ The function $\text{\hspace{0.17em}}y=\mathrm{cos}x\text{\hspace{0.17em}}$ is one-to-one on $\text{\hspace{0.17em}}\left[0,\pi \right];\text{\hspace{0.17em}}$ thus, this interval is the range of the inverse function of $\text{\hspace{0.17em}}y=\mathrm{cos}x,f\left(x\right)={\mathrm{cos}}^{-1}x.\text{\hspace{0.17em}}$

Since the functions $\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y={\mathrm{cos}}^{-1}x\text{\hspace{0.17em}}$ are inverse functions, why is $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(\mathrm{cos}\left(-\frac{\pi }{6}\right)\right)\text{\hspace{0.17em}}$ not equal to $\text{\hspace{0.17em}}-\frac{\pi }{6}?$

Explain the meaning of $\text{\hspace{0.17em}}\frac{\pi }{6}=\mathrm{arcsin}\left(0.5\right).$

$\frac{\pi }{6}\text{\hspace{0.17em}}$ is the radian measure of an angle between $\text{\hspace{0.17em}}-\frac{\pi }{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{\pi }{2}$ whose sine is 0.5.

Most calculators do not have a key to evaluate $\text{\hspace{0.17em}}{\mathrm{sec}}^{-1}\left(2\right).\text{\hspace{0.17em}}$ Explain how this can be done using the cosine function or the inverse cosine function.

Why must the domain of the sine function, $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ be restricted to $\text{\hspace{0.17em}}\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\text{\hspace{0.17em}}$ for the inverse sine function to exist?

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 $\text{\hspace{0.17em}}\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\text{\hspace{0.17em}}$ so that it is one-to-one and possesses an inverse.

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
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
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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The text passage contains a mathematical question and its answer. The question asks for a simplified expression for \(\sin(\tan^{-1}(4x))\) for \(-\frac{1}{4} \le x \le \frac{1}{4}\). The answer provided is \(\frac{4x}{\sqrt{16x^2 + 1}}\).

**Content Structure:**
The passage presents a specific mathematical problem involving inverse trigonometric functions and provides the solution immediately after the question.

**Analysis:**
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Given a text passage, determine:
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First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
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# 5.6 Inverse trigonometric functions  (Page 6/15)

 Page 6 / 15

Find a simplified expression for $\text{\hspace{0.17em}}\mathrm{sin}\left({\mathrm{tan}}^{-1}\left(4x\right)\right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}-\frac{1}{4}\le x\le \frac{1}{4}.$

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

Access this online resource for additional instruction and practice with inverse trigonometric functions.

Visit this website for additional practice questions from Learningpod.

## Key concepts

• An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
• Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
• For any trigonometric function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x={f}^{-1}\left(y\right),\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}f\left(x\right)=y.\text{\hspace{0.17em}}$ However, $\text{\hspace{0.17em}}f\left(x\right)=y\text{\hspace{0.17em}}$ only implies $\text{\hspace{0.17em}}x={f}^{-1}\left(y\right)\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is in the restricted domain of $\text{\hspace{0.17em}}f.\text{\hspace{0.17em}}$ See [link] .
• Special angles are the outputs of inverse trigonometric functions for special input values; for example, $\text{\hspace{0.17em}}\frac{\pi }{4}={\mathrm{tan}}^{-1}\left(1\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{\pi }{6}={\mathrm{sin}}^{-1}\left(\frac{1}{2}\right).$ See [link] .
• A calculator will return an angle within the restricted domain of the original trigonometric function. See [link] .
• Inverse functions allow us to find an angle when given two sides of a right triangle. See [link] .
• In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, $\text{\hspace{0.17em}}\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(x\right)\right)=\sqrt{1-{x}^{2}}.\text{\hspace{0.17em}}$ See [link] .
• If the inside function is a trigonometric function, then the only possible combinations are $\text{\hspace{0.17em}}{\mathrm{sin}}^{-1}\left(\mathrm{cos}\text{\hspace{0.17em}}x\right)=\frac{\pi }{2}-x\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}0\le x\le \pi \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(\mathrm{sin}\text{\hspace{0.17em}}x\right)=\frac{\pi }{2}-x\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}-\frac{\pi }{2}\le x\le \frac{\pi }{2}.$ See [link] and [link] .
• When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See [link] .
• When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See [link] .

## Verbal

Why do the functions $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{sin}}^{-1}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{cos}}^{-1}x\text{\hspace{0.17em}}$ have different ranges?

The function $\text{\hspace{0.17em}}y=\mathrm{sin}x\text{\hspace{0.17em}}$ is one-to-one on $\text{\hspace{0.17em}}\left[-\frac{\pi }{2},\frac{\pi }{2}\right];\text{\hspace{0.17em}}$ thus, this interval is the range of the inverse function of $\text{\hspace{0.17em}}y=\mathrm{sin}x,$ $f\left(x\right)={\mathrm{sin}}^{-1}x.\text{\hspace{0.17em}}$ The function $\text{\hspace{0.17em}}y=\mathrm{cos}x\text{\hspace{0.17em}}$ is one-to-one on $\text{\hspace{0.17em}}\left[0,\pi \right];\text{\hspace{0.17em}}$ thus, this interval is the range of the inverse function of $\text{\hspace{0.17em}}y=\mathrm{cos}x,f\left(x\right)={\mathrm{cos}}^{-1}x.\text{\hspace{0.17em}}$

Since the functions $\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y={\mathrm{cos}}^{-1}x\text{\hspace{0.17em}}$ are inverse functions, why is $\text{\hspace{0.17em}}{\mathrm{cos}}^{-1}\left(\mathrm{cos}\left(-\frac{\pi }{6}\right)\right)\text{\hspace{0.17em}}$ not equal to $\text{\hspace{0.17em}}-\frac{\pi }{6}?$

Explain the meaning of $\text{\hspace{0.17em}}\frac{\pi }{6}=\mathrm{arcsin}\left(0.5\right).$

$\frac{\pi }{6}\text{\hspace{0.17em}}$ is the radian measure of an angle between $\text{\hspace{0.17em}}-\frac{\pi }{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{\pi }{2}$ whose sine is 0.5.

Most calculators do not have a key to evaluate $\text{\hspace{0.17em}}{\mathrm{sec}}^{-1}\left(2\right).\text{\hspace{0.17em}}$ Explain how this can be done using the cosine function or the inverse cosine function.

Why must the domain of the sine function, $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ be restricted to $\text{\hspace{0.17em}}\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\text{\hspace{0.17em}}$ for the inverse sine function to exist?

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 $\text{\hspace{0.17em}}\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\text{\hspace{0.17em}}$ so that it is one-to-one and possesses an inverse.

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
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
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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Got questions? Join the online conversation and get instant answers!
user
The text passage contains a mathematical question and its answer. The question asks for a simplified expression for \(\sin(\tan^{-1}(4x))\) for \(-\frac{1}{4} \le x \le \frac{1}{4}\). The answer provided is \(\frac{4x}{\sqrt{16x^2 + 1}}\).

**Content Structure:**
The passage presents a specific mathematical problem involving inverse trigonometric functions and provides the solution immediately after the question.

**Analysis:**
TYPE: SINGLE_QUESTION_WITH_ANSWER