Inverse Trigonometric Functions

=tan⁻¹ $\left(\frac{tan\pi }{3}\right)$ − $\pi$ + sec⁻¹$\left(sec\frac{\pi }{3}\right)$

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

= $\frac{\pi -3\pi +\pi }{3}$

= $\frac{-\pi }{3}.$

Option B is correct.

Given, Sin⁻¹x=y.

(E) We know that the principal value branch of Sin⁻¹ is

$[\frac{-\pi }{2},\frac{\pi }{2}]$ Hence,  $\frac{-\pi }{2}$ ≤ y ≤ $\frac{\pi }{2}$

Option B is correct.

cos⁻¹$\frac{1}{2}$ + 2Sin⁻¹ $\frac{1}{2}$ = cos⁻¹ $\left(cos\frac{\pi }{3}\right)$ + 2*Sin⁻¹$\left(sin\frac{\pi }{6}\right)$

= $\frac{\pi }{3}+2*\frac{\pi }{6}$

= $\frac{\pi }{3}+\frac{\pi }{3}$

= $\frac{2\pi }{3}$

tan⁻¹ (1) + cos⁻¹ $\left(\frac{1}{2}\right)$ +  sin ⁻¹$\left(\frac{-1}{2}\right)$

Let cos ^-1$\left(\frac{-1}{2}\right)$ =y Then cos y = $\frac{-1}{2}$ = − cos
$\frac{\mathrm{}\pi }{3}$$\frac{}{}$ cos $\left(\pi -\frac{x}{3}\right)$

= cos $\frac{3\pi -\pi }{3}$

= cos $\frac{2\pi }{3}$

$$  (E) We know that the range of principal value

branch of cos⁻¹ is [0, $\frac{\pi }{}$] and cos $\frac{2x}{3}$ = $\frac{-1}{2}$

Principal value of cos−1 $\left(\frac{-1}{2}\right)$ is $\frac{2x}{3}$

Let sin⁻¹ $$ $\left(\frac{-1}{2}\right)$ =y. Then, sin y=- $\frac{-1}{2}$

We know that the range of principal value branch of sin⁻¹ is $-\frac{\pi }{2},\;$

and sin⁻¹ $\left(\frac{-1}{2}\right)$ =−sin⁻¹$\frac{-1}{2}$ (sin (-x) = -sin x)

= $\left(-\frac{\pi }{6}\right)$ = sin y (as sin
$\frac{\pi }{6}$
$\frac{}{}$ = $\frac{1}{2}$ )

Principal value of sin⁻¹ $\left(-\frac{1}{2}\right)$ is $\left(-\frac{\pi }{6}\right)$

students can check the table for the principal values for all ITFs below;

To understand this, Assume you have a bucket that has infinite number of apples and if your mother asks "give me the apple". How will you figure out which one is "The Apple", she is asking for.

Similarly any inversre trigonometric functions behaves like a Many-one Function; which means,

For Example $\sin^ {-1}\left (\frac {2} {3}\right)$ can have many solutions, we need to fix one solutions which can be used as standerd value for the function.

• A standerd value (Angles) of any inverse trigonometric value lies between a fiexed range is known as principal value. 

For Ex; The value of $\sin^ {-1}\left (\frac {2} {3}\right)$ will always lie between –? /2 to? /2.

$y={\sin }^{?1}\left(\frac{2}{3}\right)?\sin \left(y\right)=\frac{2}{3}?y?[?\frac{?}{2},\frac{?}{2}]$