CBSE - Class 12 Mathematics Inverse Trigonometric Functions Worksheet
1.
Prove the following: $3\cos^{-1} x = \cos^{-1} (4x^3 - 3x)$, $x \in [\frac{1}{2}, 1]$
2.
Prove that $\sin^{-1} \frac{8}{17} + \sin^{-1} \frac{3}{5} = \tan^{-1} \frac{77}{36}$
3.
Find the principal values of the following: $\cos^{-1} (\frac{-1}{\sqrt{2}})$
4.
Write the following functions in the simplest form: $\tan^{-1} (\frac{3a^2x - x^3}{a^3 - 3ax^2})$, $a > 0$; $\frac{-a}{\sqrt{3}} < x < \frac{a}{\sqrt{3}}$
5.
Find the value of the following: $\cos^{-1}(\cos(\frac{13\pi}{6}))$
6.
Find the principal values of the following: $\cos^{-1} (\frac{-1}{2})$
7.
Write the following functions in the simplest form: $\tan^{-1} (\frac{\cos x - \sin x}{\cos x + \sin x})$, $\frac{-\pi}{4} < x < \frac{3\pi}{4}$
8.
Write the following functions in the simplest form: $\tan^{-1} \frac{x}{\sqrt{a^2 - x^2}}$, $|x| < a$
9.
$\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})$ is equal to
a.
$\pi$
b.
$\frac{-\pi}{2}$
c.
$0$
d.
$2\sqrt{3}$
10.
$\sin^{-1}(1 - x) - 2\sin^{-1} x = \frac{\pi}{2}$, then $x$ is equal to
a.
$0, \frac{1}{2}$
b.
$1, \frac{1}{2}$
c.
$0$
d.
$\frac{1}{2}$
11.
Find the principal values of the following: $\tan^{-1} (-1)$
12.
Prove that $\tan^{-1} \frac{63}{16} = \sin^{-1} \frac{5}{13} + \cos^{-1} \frac{3}{5}$
13.
Find the principal values of the following: $\sec^{-1} (\frac{2}{\sqrt{3}})$
14.
If $\sin^{-1} x = y$, then
a.
$0 \leq y \leq \pi$
b.
$\frac{-\pi}{2} \leq y \leq \frac{\pi}{2}$
c.
$0 < y < \pi$
d.
$\frac{-\pi}{2} < y < \frac{\pi}{2}$
15.
Prove that $\cos^{-1} \frac{12}{13} + \sin^{-1} \frac{3}{5} = \sin^{-1} \frac{56}{65}$
16.
$\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)$ is equal to
a.
$\pi$
b.
$\frac{-\pi}{3}$
c.
$\frac{\pi}{3}$
d.
$\frac{2\pi}{3}$
17.
$\sin(\tan^{-1} x)$, $|x| < 1$ is equal to
a.
$\frac{x}{\sqrt{1-x^2}}$
b.
$\frac{1}{\sqrt{1-x^2}}$
c.
$\frac{1}{\sqrt{1+x^2}}$
d.
$\frac{x}{\sqrt{1+x^2}}$
18.
Find the principal values of the following: $\cot^{-1} (\sqrt{3})$
19.
Find the values of each of the expressions: $\tan^{-1}(\tan(\frac{3\pi}{4}))$
20.
Solve the following equations: $\tan^{-1} \frac{1-x}{1+x} = \frac{1}{2} \tan^{-1} x, (x > 0)$
