Ncert Solutions for Class 12

Inverse Trigonometric Functions

• Exercise 2.1

• Exercise 2.2

• Miscellaneous Exercise

Inverse Trigonometric Functions

Exercise 2.1

Find the principal values of the following:

1. \(sin^{-1}({-\frac{1}{2}})\)

2. \(cos^{-1}({\frac{\sqrt{3}}{2}})\)

3. \(cosec^{-1}(2)\)

4. \(tan^{-1}(-\sqrt{3})\)

5. \(cos^{-1}({-\frac{1}{2}})\)

6. \(tan^{-1}(-1)\)

7. \(sec^{-1}({\frac{2}{\sqrt 3}})\)

8. \(cot^{-1}(\sqrt 3)\)

9. \(cos^{-1}({-\frac{1}{\sqrt 2}})\)

10. \(cosec^{-1}(-\sqrt{2})\)

Find the values of the following:

11. \(tan^{-1}(1)\)+\(cos^{-1}({-\frac{1}{\sqrt 2}})\)+\(sin^{-1}({-\frac{1}{\sqrt 2}})\).

12. \(cos^{-1}({\frac{1}{\sqrt 2}})\)+2\(sin^{-1}({\frac{1}{\sqrt 2}})\).

13. If \(sin^{-1}x=y\), then

(A) \(0\le y\le\pi\)

(B) \(-\frac{\pi}{2}\le y \le -\frac{\pi}{2}\)

(C) \(0 < y <\pi\)

(D) \(-\frac{\pi}{2} < y < -\frac{\pi}{2}\) 

14. \(tan^{-1}{\sqrt{3}}-sec^{-1}(-2)\) is equal to:

(A) \(\pi\)

(B) \(-\frac{\pi}{3}\)

(C) \(\frac{\pi}{3}\)

(D) \(\frac{2\pi}{3}\)

Inverse Trigonometric Functions

Exercise 2.2

Prove the following:

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. \(tan^{-1}[2cos(2sin^{-1}{\frac{1}{2}})]\).

9. \(tan^{-1}[sin^{-1}{\frac{2x}{1+x^2}}+cos^{-1}{\frac{1-y^2}{1+y^2}}]\), \(|x|<1, y>0\) and \(xy<1\).

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. \(cos^{-1}(cos{\frac{7\pi}{6}})\).

(A) \(\frac{7\pi}{6}\)

(B) \(\frac{5\pi}{6}\)

(C) \(\frac{\pi}{3}\)

(D) \(\frac{\pi}{6}\)

14. \(sin(\frac{\pi}{3}-sin^{-1}(-\frac{1}{2}))\).

(A) \(\frac{1}{2}\)

(B) \(\frac{1}{3}\)

(C) \(\frac{1}{4}\)

(D) \(1\)

15. \(tan^{-1}{\sqrt3}-cot^{-1}({-\sqrt3})\) is equal to:

(A) \(\pi\)

(B) \(-\frac{\pi}{2}\)

(C) \(0\)

(D) \(2\sqrt3\)

Inverse Trigonometric Functions

Miscellaneous Exercise

Find the value of the following:

1. \(cos^{-1}(cos\frac{13\pi}{6})\).

2. \(tan^{-1}(tan\frac{7\pi}{6})\).

3.  \(2sin^{-1}{\frac{3}{5}}\)\(=tan^{-1}{\frac{24}{7}}\).

4. \(sin^{-1}{\frac{8}{17}}+sin^{-1}{\frac{3}{5}}\)\(=tan^{-1}{\frac{77}{36}}\).

5. \(cos^{-1}{\frac{4}{5}}+cos^{-1}{\frac{12}{13}}\)\(=cos^{-1}{\frac{33}{65}}\).

6. \(cos^{-1}{\frac{12}{13}}+sin^{-1}{\frac{3}{5}}\)\(=sin^{-1}{\frac{56}{65}}\).

7. \(tan^{-1}{\frac{63}{16}}=sin^{-1}{\frac{5}{13}}\)\(+cos^{-1}{\frac{3}{5}}\).

Prove that:

8. \(tan^{-1}{\sqrt{x}}=\frac{1}{2}cos^{-1}{\frac{1-x}{1+x}}\).

9. \(cot^{-1}(\frac{\sqrt{1+sinx}+\sqrt{1-sinx}}{\sqrt{1+sinx}+\sqrt{1-sinx}})\)\(=\frac{x}{2},\;x\in(0,\frac{\pi}{4})\).

10. \(tan^{-1}(\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}})\)\(=\frac{\pi}{4}-\frac{1}{2}cos^{-1}x\),\(\;-\frac{1}{\sqrt2}\le{x}\le{1}\).

Solve the following equations:

11. \(2tan^{-1}(cosx)=tan^{-1}(2cosecx)\).

12. \(tan^{-1}{\frac{1-x}{1+x}}\)\(=\frac{1}{2}tan^{-1}x,\;(x>0)\)

13. \(sin(tan^{-1}x),|x|<1\) 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) \(\frac{x}{\sqrt{1+x^2}}\)

14. \(sin^{-1}(1-x)-2sin^{-1}x\)\(=\frac{\pi}{2}\), then x is equal to:

(A) \(0,\frac{1}{2}\)

(B) \(1,\frac{1}{2}\)

(C) \(0\)

(D) \(\frac{1}{2}\)