 # Chapter 2: Inverse Trigonometric Functions EXERCISE 2.1

In this page we have NCERT Solutions for Class 12 Maths Chapter 2: Inverse Trigonometric Functions for EXERCISE 2.1 . Hope you like them and do not forget to like , social share and comment at the end of the page.

Formula's used
Principle Value branch of Trigonometric Inverse Functions along with domain and range ### Find the principal values of the following

Question 1. $\sin^{-1}\left ( -\frac{1}{2} \right )$
Solution
Let $\sin^{-1}\left ( -\frac{1}{2} \right )$ = x, then
$\sin {x} = -\frac{1}{2} = - \sin\frac{ \pi}{6} = \sin ( - \frac{ \pi}{6} )$
We know from above table that
The principal value branch range for sin-1 is $\left [ -\frac{ \pi}{2}, \frac{ \pi}{2} \right ]$ and $\sin ( -\frac{ \pi}{6} ) = - \frac{1}{2}$
Therefore principal value for $\sin^{-1}\left ( -\frac{1}{2} \right ) \; is \; - \frac{ \pi}{6}$

Question 2. $\cos^{-1}\left ( - \frac{\sqrt{3}}{2} \right )$
Solution
Let $\cos^{-1}\left ( - \frac{\sqrt{3}}{2} \right )$ = x, then
$\cos x = \frac{\sqrt{3}}{2} = \cos (\frac{\pi}{6})$ We know from above table that
The principal value branch range for cos-1 is $\left [ 0 , \pi \right ]$ and $\cos (\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$
Therefore, principal value for $\cos^{-1}\left ( - \frac{\sqrt{3}}{2} \right ) \; is \; \frac{\pi}{6}$

Question 3. cosec-1 (2)
Solution
Let cosec-1 (2) = x. Then, cosec x = 2 = cosec $(\frac{\pi}{6} )$
We know from above table that
The principal value branch range for cosec-1 is $\left [ -\frac{\pi}{2}, \frac{\pi}{2}\right ] - {0}$ and cosec$(\frac{\pi}{6} )$ = 2
Therefore, principal value for cosec-1 (2) is $\frac{\pi}{6}$

Question 4. $\tan^{-1} \left ( - \sqrt{3} \right )$
Solution

Let $\tan^{-1} \left ( - \sqrt{3} \right ) = x$
Then, $\tan = - \sqrt{3} = - \tan \frac{\pi}{3} \tan (- \frac{\pi}{3})$
We know from above table that
The principal value branch range for $\tan^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ] \; and \; \tan\left ( -\frac{\pi}{3} \right ) = -\sqrt{3}$
Therefore, principal value for $\tan^{-1} \left ( - \sqrt{3} \right ) \; is \; -\frac{\pi}{3}$

Question 5. $\cos^{-1}\left ( -\frac{1}{2} \right )$
Solution
Let $\cos^{-1}\left ( -\frac{1}{2} \right )$ = x,
Then $\cos x = -\frac{1}{2} = -cos \frac{\pi}{3} = \cos ( \pi - \frac{\pi}{3} ) = \cos( \frac{2 \pi}{3} )$
We know from above table that
The principal value branch range for $\cos ^{-1} \; is \; \left [ 0 , \pi \right ] \; and \; \cos \left ( \frac{2 \pi}{3} \right ) = - \frac{1}{2}$
Therefore, principal value for $\cos^{-1}\left ( -\frac{1}{2} \right ) \; is \; \frac{2 \pi}{3}$

Question 6. $\tan^{-1} (-1)$
Solution

Let $\tan^{-1} (-1) = x$,
Then, tan x = -1 = $-\tan ( \frac{\pi}{4} ) = \tan ( - \frac{\pi}{4} )$
We know from above table that
The principal value branch range for $\tan^{-1} \; is \; \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right ) \; and \; \tan ( - \frac{\pi}{4} ) = -1$
Therefore, principal value for $\tan^{-1} (-1) \; is \; -\frac{\pi}{4}$

Question 7. $\sec^{-1} \left ( \frac{2}{\sqrt{3}} \right )$
Solution

Let $\sec^{-1} \left ( \frac{2}{\sqrt{3}} \right ) = x$,
Then $\sec x = \frac{2}{\sqrt{3}} = \sec (\frac{\pi}{6})$
We know from above table that
The principal value branch range for $\sec^{-1} \; is \; \left [ 0 , \pi \right ] - \left \{ \frac{ \pi }{2} \right \} \; and \; \sec (\frac{\pi}{6}) = \frac{2}{\sqrt{3}}$
Therefore, principal value for $\sec^{-1} \left ( \frac{2}{\sqrt{3}} \right ) \; is ; \frac{\pi}{6}$

Question 8. $\cot^{-1} \sqrt{3}$
Solution
Let $\cot^{-1} \sqrt{3} = x$,
Then $\cot x = \sqrt {3} = \cot \left ( \frac{\pi}{6} \right )$
We know from above table that
The principal value branch range for cot-1 is $( 0 , \pi )$ and $\cot \left ( \frac{\pi}{6} \right ) = \sqrt{3}$
Therefore, principal value for $\cot^{-1} \sqrt{3} = \frac{\pi}{6}$

Question 9. Find principal value for $\cos^{-1} \left ( - \frac{1}{\sqrt{2}} \right )$
Solution
Let $\cos^{-1} \left ( - \frac{1}{\sqrt{2}} \right ) = a$
Then $\cos x = \frac{-1}{\sqrt{2}} = - \cos \left ( \frac{\pi}{4} \right ) = \cos \left ( \pi - \frac{\pi}{4} \right ) = \cos \left ( \frac{3 \pi}{4} \right )$
We know from above table that
The principal value branch range for cos-1 is $[0 , \pi] \; and \; \cos \left ( \frac{3 \pi}{4} \right ) = -\frac{1}{\sqrt{2}}$
Therefore, principal value for $\cos^{-1} \left ( - \frac{1}{\sqrt{2}} \right ) \; is \; \frac{3 \pi }{4}$

Question 10. Find principal value for cosec-1 $\left ( -\sqrt{2} \right )$
Solution
Let cosec-1$\left ( -\sqrt{2} \right )$ = x, Then
cosec x = $-\sqrt{2}$ = -cosec$\left ( \frac{\pi}{4}\right )$ = cosec $\left ( -\frac{\pi}{4}\right )$
We know,
The principal value branch range for cosec-1 is $\left [ -\frac{\pi}{2} , \frac{\pi}{2} \right ] - \left \{ 0 \right \}$ and cosec$\frac{-\pi}{4} = -\sqrt{2}$
Therefore, principal value for cosec-1 $\left ( -\sqrt{2} \right ) \; is \; -\frac{\pi}{4}$

### Find the values of the following

Question 11. Solve $\tan ^{-1}(1) + \cos^{-1} \left ( -\frac{1}{2} \right ) + \sin ^{-1}\left ( -\frac{1}{2} \right )$
Solution

Let $\tan ^{-1}(1) = x$, then
$\tan x = 1 = \tan \frac{\pi}{4}$ We know from above table that
The principal value branch range for $\tan ^{-1} \; is \; \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )$
$\tan ^{-1}(1) = \frac{\pi}{4}$
Let $\cos^{-1} \left ( -\frac{1}{2} \right ) = y$, then
$\cos y = -\frac{1}{2} = -\cos \frac {\pi}{3} = \cos\left ( \pi - \frac{\pi}{3} \right ) = \cos\left ( \frac{2\pi}{3} \right )$
We know from above table that
The principal value branch range for cos-1 is $[0 , \pi]$
$\cos ^{-1} \left ( -\frac{1}{2} \right ) = \frac{2 \pi }{3}$
Let $\sin^{-1}\left ( -\frac{1}{2} \right ) = z$, then
$\sin z = - \frac{1}{2} = - \sin \frac{\pi}{6} = \sin \left ( -\frac{\pi}{6} \right )$
We know from above table that
The principal value branch range for $\sin ^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$
$\sin^{-1} \left ( -\frac{1}{2} \right ) = - \frac{\pi}{6}$ Now
$\tan ^{-1}(1) + \cos^{-1} \left ( -\frac{1}{2} \right ) + \sin ^{-1}\left ( -\frac{1}{2} \right )$ $= \frac{\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{6} = \frac{3\pi}{4}$
Question 12. Solve $\cos ^{-1} \left ( \frac{1}{2} \right ) + 2 \sin ^{-1} \left ( \frac{1}{2} \right )$
Solution
Let $\cos ^{-1} \left ( \frac{1}{2} \right ) = x$, then $\cos x = \frac{1}{2} = \cos \frac{\pi}{3}$
We know from above table that
The principal value branch range for cos-1 is $\left [0 , \pi \right ]$
$\cos ^{-1} \left ( \frac{1}{2} \right ) = \frac{\pi}{3}$
Let $\sin ^{-1} \left (- \frac{1}{2} \right ) = y$, then $\sin y = \frac{1}{2} = \sin \frac{\pi}{6}$
We know,
The principal value branch range for $\sin ^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$
$\sin ^{-1} \left ( \frac{1}{2} \right ) = \frac{\pi}{6}$
Now,
$\cos ^{-1} \left ( \frac{1}{2} \right ) + 2 \sin ^{-1} \left ( \frac{1}{2} \right )$ $= \frac{\pi}{3} + 2 \times \frac{\pi}{6} = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3}$

### Multiple Choice Questions

Question 13. If sin-1 a = b, then
(i) $0 \leq b \leq \pi$
(ii) $-\frac{\pi}{2} \leq b \leq \frac{\pi}{2}$
(iii) $0 < b < \pi$
(iv) $-\frac{\pi}{2} < b < \frac{\pi}{2}$
Solution
Given sin-1 a = b
We know from above table that
The principal value branch range for $\sin ^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ]$
Therefore, $-\frac{\pi}{2} \leq b \leq \frac{\pi}{2}$

Question 14. The value of $\tan ^{-1} \sqrt{3} - \sec ^{-1}(-2)$ is
(i) $\pi$
(ii) $- \frac{\pi}{3}$
(iii)$\frac{\pi}{3}$
(iv) $\frac{2 \pi}{3}$
Solution
Let $\tan ^{-1} \sqrt{3} = x$, then
$\tan x = \sqrt{3} = \tan \frac{\pi}{3}$
We know from above table that
The principal value branch range for $\tan ^{-1} \; is \; \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )$
$\tan ^{-1}\sqrt{3} = \frac{\pi}{3}$
Let sec-1(-2) = y, then
sec y = -2 = $- \sec \frac{\pi}{3} = \sec \left ( \pi - \frac{\pi}{3} \right ) = \sec \left ( \frac{2 \pi}{3} \right )$
We know from above table that
The principal value branch range for sec-1 is $[0 , \pi] - \left \{ \frac{\pi}{2} \right \}$
$\sec ^{-1}(-2) = \frac{2 \pi}{3}$
Now,
$\tan ^{-1} \sqrt{3} - \sec ^{-1}(-2) = \frac{\pi}{3} - \frac{2 \pi}{3} = - \frac{\pi}{3}$
Hence option (ii) is correct