Inverse Trigonometric Functions: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Principal value of ${\csc }^{-1}(-2)$.
2. Principal value of ${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$.
3. Value of ${\tan }^{-1}\left(\tan \frac{5\pi }{4}\right)$.
4. Range of ${\tan }^{-1}x$.
5. Value of $\sin \left(\frac{\pi }{2}-{\sin }^{-1}\frac{1}{2}\right)$.
6. Domain of ${\cos }^{-1}(x-1)$.
7. Value of $\cot ({\tan }^{-1}3)$.
8. Value of ${\sin }^{-1}1+{\cos }^{-1}1$.
9. Name a branch of ${\sin }^{-1}$ other than the principal one.
10. Value of $\sec ({\tan }^{-1}1)$.

D. Short answer I (2 marks each)

1. Evaluate ${\sin }^{-1}\left(\sin \frac{4\pi }{5}\right)$.
2. Find the domain of ${\sin }^{-1}(2x-1)$.
3. Evaluate ${\tan }^{-1}\left[2\sin \left(2{\cos }^{-1}\frac{\sqrt{3}}{2}\right)\right]$.
4. Simplify ${\cos }^{-1}\frac{1-x^2}{1+x^2}$, $0<x<1$.
5. Find $\cos \left({\tan }^{-1}\frac{3}{4}\right)$.
6. Show that ${\sin }^{-1}\frac{2x}{1+x^2}=2{\tan }^{-1}x$ for $|x|\le 1$.
7. Evaluate ${\cos }^{-1}\left(\cos \frac{4\pi }{3}\right)$.
8. Find the range of $3{\sin }^{-1}x+\frac{\pi }{2}$.

Rough work

E. Short answer II (3 marks each)

1. Write ${\tan }^{-1}\frac{\sqrt{1+x^2}+1}{x}$, $x\ne 0$, in the simplest form.
2. Prove ${\tan }^{-1}\frac{1}{2}+{\tan }^{-1}\frac{1}{3}=\frac{\pi }{4}$.
3. Find the value of $\sin \left(2{\tan }^{-1}\frac{1}{3}\right)+\cos ({\tan }^{-1}2\sqrt{2})$.
4. Draw the graph of $y={\cos }^{-1}x$ and state its domain and range.
5. Prove ${\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}={\sin }^{-1}\frac{63}{65}$.
6. Solve ${\tan }^{-1}2x+{\tan }^{-1}3x=\frac{\pi }{4}$.

F. Long answer (5 marks each)

1. (a) Prove ${\cot }^{-1}\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}=\frac{x}{2}$ for $x\in (0,\frac{\pi }{4})$. (b) Find the domain and range of $f(x)={\sin }^{-1}\sqrt{x-1}$.
2. (a) Simplify ${\tan }^{-1}\frac{a\cos x-b\sin x}{b\cos x+a\sin x}$ given $\frac{a}{b}\tan x>-1$. (b) Find the principal values of ${\cot }^{-1}(-\sqrt{3})$, ${\sec }^{-1}(-2)$, ${\csc }^{-1}\frac{2}{\sqrt{3}}$.

G. HOTS (higher-order thinking)

1. Why must the domain of a trigonometric function be restricted before inverting it?
2. Is ${\sin }^{-1}x+{\sin }^{-1}x={\sin }^{-1}2x$? Test with $x=1$.
3. For which $x$ is ${\cos }^{-1}(\cos x)=x$?
4. What is the largest value of ${\tan }^{-1}x+{\cot }^{-1}x$?

20 MCQs

1. ${\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
   (a) $\frac{4\pi }{3}$ (b) $\frac{2\pi }{3}$ (c) $-\frac{\pi }{3}$ (d) $\frac{\pi }{3}$
2. Range of ${\cot }^{-1}x$
   (a) $(0,\pi )$ (b) $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ (c) $[0,\pi ]$ (d) $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
3. ${\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)$
   (a) $-\frac{\pi }{4}$ (b) $\frac{\pi }{4}$ (c) $\frac{5\pi }{4}$ (d) $\frac{3\pi }{4}$
4. Domain of ${\cos }^{-1}(3x)$
   (a) $[-1,1]$ (b) $\left[-\frac{1}{3},\frac{1}{3}\right]$ (c) $[-3,3]$ (d) R
5. ${\tan }^{-1}\sqrt{3}-{\sec }^{-1}(-2)$
   (a) $\pi$ (b) $-\frac{\pi }{3}$ (c) $\frac{\pi }{3}$ (d) $\frac{2\pi }{3}$
6. ${\sin }^{-1}\left(\sin \frac{3\pi }{5}\right)$
   (a) $\frac{3\pi }{5}$ (b) $\frac{\pi }{5}$ (c) $-\frac{2\pi }{5}$ (d) $\frac{2\pi }{5}$
7. ${\tan }^{-1}x+{\cot }^{-1}x$
   (a) $\frac{\pi }{2}$ (b) $\pi$ (c) 0 (d) $\frac{\pi }{4}$
8. $\cos \left({\tan }^{-1}\frac{3}{4}\right)$
   (a) $\frac{3}{5}$ (b) $\frac{3}{4}$ (c) $\frac{4}{5}$ (d) $\frac{5}{3}$
9. ${\sec }^{-1}2+{\csc }^{-1}2$
   (a) $\pi$ (b) $\frac{2\pi }{3}$ (c) $\frac{\pi }{3}$ (d) $\frac{\pi }{2}$
10. ${\cot }^{-1}(-\sqrt{3})$
    (a) $-\frac{\pi }{6}$ (b) $\frac{5\pi }{6}$ (c) $\frac{\pi }{6}$ (d) $\frac{2\pi }{3}$
11. $\sin \left({\cos }^{-1}\frac{4}{5}\right)$
    (a) $\frac{4}{5}$ (b) $\frac{4}{3}$ (c) $\frac{3}{5}$ (d) $\frac{3}{4}$
12. If ${\cos }^{-1}x=\frac{\pi }{3}$ then $x=$
    (a) $\frac{1}{2}$ (b) $\frac{\sqrt{3}}{2}$ (c) $-\frac{1}{2}$ (d) 1
13. Which is not defined?
    (a) ${\sin }^{-1}1$ (b) ${\sec }^{-1}\frac{1}{2}$ (c) ${\tan }^{-1}100$ (d) ${\cos }^{-1}0$
14. ${\tan }^{-1}1+{\tan }^{-1}(-1)$
    (a) $\frac{\pi }{2}$ (b) $\pi$ (c) 0 (d) $\frac{\pi }{4}$
15. $2{\tan }^{-1}\frac{1}{2}=$
    (a) ${\tan }^{-1}1$ (b) ${\tan }^{-1}\frac{3}{4}$ (c) ${\tan }^{-1}\frac{2}{3}$ (d) ${\tan }^{-1}\frac{4}{3}$
16. Range of ${\sec }^{-1}x$
    (a) $[0,\pi ]-\{\frac{\pi }{2}\}$ (b) $[0,\pi ]$ (c) $(0,\pi )$ (d) $[-\frac{\pi }{2},\frac{\pi }{2}]$
17. ${\cos }^{-1}(\cos 2\pi )$
    (a) 0 (b) $2\pi$ (c) $\pi$ (d) $-2\pi$
18. ${\sin }^{-1}\frac{1}{2}+{\cos }^{-1}\frac{1}{2}$
    (a) $\pi$ (b) $\frac{\pi }{6}$ (c) $\frac{\pi }{3}$ (d) $\frac{\pi }{2}$
19. $\tan \left({\sin }^{-1}\frac{3}{5}\right)$
    (a) $\frac{4}{5}$ (b) $\frac{3}{4}$ (c) $\frac{4}{3}$ (d) $\frac{3}{5}$
20. Largest value in the range of ${\sin }^{-1}x$
    (a) $\pi$ (b) 1 (c) $\frac{\pi }{2}$ (d) $\frac{3\pi }{2}$

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: ${\cos }^{-1}\left(-\frac{1}{2}\right)=\frac{2\pi }{3}$. R: ${\cos }^{-1}(-x)=\pi -{\cos }^{-1}x$.
2. A: ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)=\frac{2\pi }{3}$. R: ${\sin }^{-1}(\sin x)=x$ for $x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$.
3. A: The domain of ${\sin }^{-1}(x^2)$ is $[-1,1]$. R: $0\le x^2\le 1$ for $|x|\le 1$.
4. A: ${\tan }^{-1}x$ takes the value $\frac{\pi }{2}$. R: the range of ${\tan }^{-1}$ is $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$.
5. A: ${\sec }^{-1}(0.5)$ is not defined. R: ${\sec }^{-1}x$ is defined only for $|x|\ge 1$.
6. A: ${\tan }^{-1}2+{\tan }^{-1}3=\frac{3\pi }{4}$. R: ${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\frac{x+y}{1-xy}$ for all $x,y$.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (d) · 5 (a) · 6 (c)

Answers to the practice set