Class 12
Inverse Trigonometric Functions
Exercise 2.2
1. \(3sin^{-1}{x}=sin^{-1}(3x-4x^3)\), \(x\in[-\frac{1}{2},\frac{1}{2}]\).
2. \(3cos^{-1}{x}=cos^{-1}(4x^3-3x)\), \(x\in[\frac{1}{2},1]\).
Write the following functions in the simplest form:
3. \(tan^{-1}{\frac{\sqrt{1+x^2}-1}{x}}\), \(x\ne {0}\).
4. \(tan^{-1}(\sqrt{\frac{1-cosx}{1+cosx}})\),\(0<x<\pi\).
5. \(tan^{-1}(\frac{cosx-sinx}{cosx+sinx})\), \(-\frac{\pi}{4}<x<\frac{3\pi}{4}\).
6. \(tan^{-1}{\frac{x}{\sqrt{a^2-x^2}}}\), \(|x|<a\).
7. \(tan^{-1}(\frac{3a^2x-x^3}{a^3-3ax^2})\), \(a>0;-\frac{a}{\sqrt3}<x<\frac{a}{\sqrt3}\).
Find the values of each of the following:
8. \(tan^{-1}[2cos(2sin^{-1}{\frac{1}{2}})]\).
Find the values of each of the expressions in Exercises 10 to 15.
10. \(sin^{-1}(sin{\frac{2\pi}{3}})\).
11. \(tan^{-1}(tan{\frac{3\pi}{4}})\).
12. \(tan(sin^{-1}{\frac{3}{5}}+cot^{-1}{\frac{3}{2}})\).
13. \(cos^{-1}(cos{\frac{7\pi}{6}})\).
14. \(sin(\frac{\pi}{3}-sin^{-1}(-\frac{1}{2}))\).