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Principal Values of the Inverse Trigonometric Functions

In this page we are going to discuss about principal values of the inverse trigonometric functions .The inverse trigonometric functions are basically the reverse of the given trigonometric functions ..

These are : 

$sin^{-1}\ x$, $cos^{-1}\ x$, $tan^{-1}\ x$, $cosec^{-1}\ x$, $sec^{-1}\ x$, $cot^{-1}\ x$

Principal value is the set of all values that a function will take for different values of $x$.Set of all $x$ values is called as domain.the range of trigonometric function always lies between certain values,it does not go to infinity.here we consider the domain from $-1$ to $1$.

The principal values or ranges of different inverse functions are as under :

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Principal Values of the Inverse Trigonometric Functions Table

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Function Domain Principal Value (Range)
$y$ = $sin^{-1}\ x$
[- 1, 1] [-$\frac{\pi}{2}$ , $\frac{\pi}{2}$]
$y$ = $cos^{-1}\ x$
[- 1, 1] [$0$, $\pi$ ]
$y$ = $tan^{-1}\ x$
R (-$\frac{\pi}{2}$ , $\frac{\pi}{2}$)
$y$ = $cosec^{-1}\ x$
R - (- 1, 1) [-$\frac{\pi}{2}$, $\frac{\pi}{2}$ ] - {$0$}
$y$ = $sec^{-1}\ x$
R - [- 1, 1]
[$0,\ \pi$] - {$\frac{\pi}{2}$}
$y$ = $cot^{-1}\ x$
R ($0$, $\pi$ )

Examples on Principal Values

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Below are the examples on principal values -

Example 1:

Calculate the principal value for the inverse function $sin^{-1}$ ($-\frac{1}{2}$) by using the table .

Solution:

We know that principal value of $sin^{-1}$ is [-$\frac{\pi}{2}$ , $\frac{\pi}{2}$]


Let $sin^{-1}$ (-$\frac{1}{2}$) = $\theta$. Then,


$sin$ $\theta$ = $\frac{1}{2}$ = $sin$ (-$\frac{\pi}{6}$)


Then


$\theta$ = $\frac{\pi}{6}$ $\theta$ [- $\frac{\pi}{2}$, $\frac{\pi}{2}$].


The principal value of $sin^{-1}$ (-$\frac{1}{2}$) is - $\frac{\pi}{6}$

Example 2:

Calculate the principal value for the inverse function $tan^{-1}\ (\sqrt{3})$ by using table .

Solution:

We know that principal - value of $tan^{-1}$ is [- $\frac{\pi}{2}$, $\frac{\pi}{2}$]


Let $tan^{-1}\ (\sqrt{3})$ = $\theta$. Then,


tan $\theta$ = $\sqrt{3}$ = $tan$ $\frac{\pi}{3}$


Then,


$\theta$ = $\frac{\pi}{3}$ $\in$ [- $\frac{\pi}{2}$, $\frac{\pi}{2}$]


The principal value of given inverse function $tan^{-1} (\sqrt{3})$ is $\frac{\pi}{3}$

Practice Problems

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Problem 1:

Calculate the principal value for the inverse function $sin^{-1}$($\sqrt{\frac{3}{2}}$) by using the table .

Problem 2:

Calculate the principal value for the inverse function $cos^{-1}$ ($\frac{1}{2}$) by using the table .

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