Find a simplified expression for
$\text{\hspace{0.17em}}\mathrm{sin}\left({\mathrm{tan}}^{-1}\left(4x\right)\right)\text{\hspace{0.17em}}$ for
$\text{\hspace{0.17em}}-\frac{1}{4}\le x\le \frac{1}{4}.$
$\frac{4x}{\sqrt{16{x}^{2}+1}}$
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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
$\text{\hspace{0.17em}}f(x),\text{\hspace{0.17em}}$ if
$\text{\hspace{0.17em}}x={f}^{-1}(y),\text{\hspace{0.17em}}$ then
$\text{\hspace{0.17em}}f(x)=y.\text{\hspace{0.17em}}$ However,
$\text{\hspace{0.17em}}f(x)=y\text{\hspace{0.17em}}$ only implies
$\text{\hspace{0.17em}}x={f}^{-1}(y)\text{\hspace{0.17em}}$ 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,
$\text{\hspace{0.17em}}\frac{\pi}{4}={\mathrm{tan}}^{-1}(1)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\frac{\pi}{6}={\mathrm{sin}}^{-1}\left(\frac{1}{2}\right).$ 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] .
Section exercises
Verbal
Why do the functions
$\text{\hspace{0.17em}}f(x)={\mathrm{sin}}^{-1}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}g(x)={\mathrm{cos}}^{-1}x\text{\hspace{0.17em}}$ 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(x)={\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
$\text{\hspace{0.17em}}y=\mathrm{cos}x,f(x)={\mathrm{cos}}^{-1}x.\text{\hspace{0.17em}}$
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.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
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
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?