Class 12 Inverse Trigonometric Functions Mock Test (CBSE)

Topics in Inverse Trigonometric Functions

• Inverse of trigonometric functions
• Domain and principal value branch
• Graphs of inverse trigonometric functions
• Properties of inverse trigonometric functions

Inverse Trigonometric Functions practice questions with answers

1. Q1Multiple choice·1 mark
   The value of $\sin^{-1}\left(\cos\left(\frac{3\pi}{5}\right)\right)$ is:

   1. (a)$\frac{\pi}{10}$
   2. (b)$\frac{7\pi}{10}$
   3. (c)$-\frac{\pi}{10}$
   4. (d)$\frac{3\pi}{5}$
   Show answer and step marks

   1. Correct option: (c) $-\frac{\pi}{10}$+1

   Answer: (c)
2. Q2Multiple choice·1 mark
   The principal value of $\cos^{-1}\left(-\frac{1}{2}\right)$ is:

   1. (a)$-\frac{2\pi}{3}$
   2. (b)$\frac{\pi}{3}$
   3. (c)$\frac{2\pi}{3}$
   4. (d)$-\frac{\pi}{3}$
   Show answer and step marks

   1. Correct option: (c) $\frac{2\pi}{3}$+1

   Answer: (c)
3. Q3Short answer·3 marks
   Prove that $\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{8}\right) = \frac{\pi}{4}$.
   Show answer and step marks

   1. Apply the property $\tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right)$ for the first two terms.+1
   2. Simplify the combination: $\tan^{-1}\left(\frac{\frac{1}{2} + \frac{1}{5}}{1 - \frac{1}{2} \times \frac{1}{5}}\right) = \tan^{-1}\left(\frac{\frac{7}{10}}{\frac{9}{10}}\right) = \tan^{-1}\left(\frac{7}{9}\right)$.+1
   3. Apply the property again with $\tan^{-1}\left(\frac{7}{9}\right)$ and $\tan^{-1}\left(\frac{1}{8}\right)$ to get $\tan^{-1}\left(\frac{\frac{7}{9} + \frac{1}{8}}{1 - \frac{7}{72}}\right) = \tan^{-1}(1) = \frac{\pi}{4}$. Hence proved.+1

   Answer: $\tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{5} = \tan^{-1}\frac{7}{9}$, and $\tan^{-1}\frac{7}{9} + \tan^{-1}\frac{1}{8} = \tan^{-1}\frac{65/72}{65/72} = \tan^{-1} 1 = \frac{\pi}{4}$.

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