Inverse Trigonometric Functions Part - 1

Questions Discussed:

Evaluate each of the following:

Question 1. { \displaystyle {{\sin }^{-1}}\left( -\frac{\sqrt{3}}{2} \right) }

Question 2. { \displaystyle {{\cot }^{-1}}\left( \frac{-1}{\sqrt{3}} \right) }

Question 3. { \displaystyle {{\tan }^{-1}}\left( -\frac{1}{\sqrt{3}} \right) }

Question 4. { \displaystyle {{\tan }^{-1}}(1)}+{ \displaystyle{{\cos }^{-1}}\left( -\frac{1}{2} \right)}+{ \displaystyle{{\sin }^{-1}}\left( -\frac{1}{2} \right) }

Question 5. { \displaystyle {{\cos }^{-1}}\left( \frac{1}{2} \right)}+{ \displaystyle2{{\sin }^{-1}}\left( \frac{1}{2} \right) }

Question 6. { \displaystyle {{\tan }^{-1}}\sqrt{3}-{{\sec }^{-1}}(-2) }

Question 7. { \displaystyle {{\sin }^{-1}}\left( \sin \frac{4\pi }{5} \right) }

Question 8. { \displaystyle {{\sin }^{-1}}\left( \sin \frac{2\pi }{3} \right) }

Question 9. { \displaystyle {{\tan }^{-1}}\left( \tan \frac{3\pi }{4} \right) }

Question 10 { \displaystyle {{\cos }^{-1}}\left( \cos \frac{7\pi }{6} \right) }

Question 11 { \displaystyle \sin \left( \frac{\pi }{3}-{{\sin }^{-1}}\left( -\frac{1}{2} \right) \right) }

Question 12 { \displaystyle {{\tan }^{-1}}\sqrt{3}-{{\cot }^{-1}}(-\sqrt{3}) }

Question 13 { \displaystyle {{\csc }^{-1}}(-2) }

Question 14 { \displaystyle {{\sin }^{-1}}\left( \sin \frac{3\pi }{5} \right) }

Question 16 { \displaystyle {{\cos }^{-1}}\left( \cos \frac{13\pi }{6} \right) }

Question 17 { \displaystyle {{\tan }^{-1}}\left( \tan \frac{7\pi }{6} \right) }

Question 18 { \displaystyle {{\cos }^{-1}}[\cos (-680{}^\circ )] }

Question 20 { \displaystyle {{\tan }^{-1}}\left( \tan \frac{5\pi }{6} \right) }

Questions Discussed:

Evaluate each of the following:

Question 15 { \displaystyle \tan \left[ \frac{1}{2}{{\cos }^{-1}}\left( \frac{2}{\sqrt{5}} \right) \right] }

Question 21. { \displaystyle {{\sin }^{-1}}\left( \cos \left( \frac{43\pi }{5} \right) \right) }

Question 22. { \displaystyle {{\sin }^{-1}}\left( -\frac{\sqrt{3}}{2} \right)}+{ \displaystyle{{\cos }^{-1}}\left( -\frac{1}{2} \right)}+{ \displaystyle{{\tan }^{-1}}\left( -\frac{1}{\sqrt{3}} \right) }

Question 23. { \displaystyle {{\tan }^{2}}({{\sec }^{-1}}2)}+{ \displaystyle{{\cot }^{2}}(\text{cose}{{\text{c}}^{-1}}3) }

Question 26. { \displaystyle \sin ({{\tan }^{-1}}x+{{\cot }^{-1}}x) }

Question 35. { \displaystyle \cot ({{\tan }^{-1}}a+{{\cot }^{-1}}a) }

Questions Discussed:

Evaluate each of the following:

Question 19 { \displaystyle \tan \left\{ 2{{\tan }^{-1}}\frac{1}{5}-\frac{\pi }{4} \right\} }

Question 24. { \displaystyle \sin \left( 2{{\tan }^{-1}}\frac{1}{3} \right)}+{ \displaystyle\cos ({{\tan }^{-1}}2\sqrt{2}) }

Question 27. { \displaystyle \sin \left( {{\cos }^{-1}}\frac{4}{5} \right) }

Question 28. { \displaystyle \sin \left( {{\cot }^{-1}}\frac{4}{3} \right) }

Question 29. { \displaystyle \sin ({{\cot }^{-1}}x) }

Question 30. { \displaystyle \cos ({{\tan }^{-1}}x) }

Question 31. { \displaystyle {{\sin }^{-1}}\frac{1}{2}-2{{\sin }^{-1}}\frac{1}{\sqrt{2}} }

Question 32. { \displaystyle \tan \left( \cos^{-1}\frac{4}{5}+\tan^{-1}\frac{2}{3} \right) }

Question 33. { \displaystyle \tan \left( 2{{\tan }^{-1}}\frac{1}{5} \right) }

Question 34. { \displaystyle \tan \left[ 2\cos \left( 2{{\sin }^{-1}}\frac{1}{2} \right) \right] }

Question 35. { \displaystyle \cot ({{\tan }^{-1}}a+{{\cot }^{-1}}a) }

Question 36. \tan { \displaystyle \frac{1}{2} } \left[ { \displaystyle \sin^{-1} \frac{2x}{1+x^2} } + { \displaystyle \cos^{-1} \frac{1-y^2}{1+y^2}} \right ], |x|<1, \, y>0 \,\text{and}\,xy < 1

Question 37. { \displaystyle \sin ({{\tan }^{-1}}x),\,\,|x| < 1 }

Question 38. \tan^{-1} { \displaystyle \left( \frac{x}{y} \right ) } - \tan^{-1} { \displaystyle \left( \frac{x-y}{x+y} \right) }

Questions Discussed:

Simplify:

Question 1. { \displaystyle {{\tan }^{-1}}\left( \sqrt{\frac{1-\cos x}{1+\cos x}} \right)},\,\,x < \pi

Question 2. { \displaystyle {{\tan }^{-1}}\left( \frac{\cos x-\sin x}{\cos x+\sin x} \right)},\,x < \pi

Question 3. { \displaystyle {{\cot }^{-1}}\left( \frac{1}{\sqrt{{{x}^{2}}-1}} \right)},\,\,|x|\,>1

Question 4. { \displaystyle {{\tan }^{-1}}\frac{1}{\sqrt{{{x}^{2}}-1}}},\,|x|\,>1

Question 5. { \displaystyle {{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)},\,\,x\ne 0

Question 6. { \displaystyle {{\tan }^{-1}}\frac{x}{\sqrt{{{a}^{2}}-{{x}^{2}}}}},\,\,|x| < a

Questions Discussed:

Simplify:

Question 7. { \displaystyle {{\tan }^{-1}}\left( \frac{3{{a}^{2}}x-{{x}^{3}}}{{{a}^{3}}-3a{{x}^{2}}} \right)},\,a>0;\,\frac{-a}{\sqrt{3}}\le x\le \frac{a}{\sqrt{3}}

Question 8. { \displaystyle {{\tan }^{-1}}\left( \frac{\cos x}{1+\sin x} \right) }

Question 9. { \displaystyle {{\tan }^{-1}}\left( \frac{\sin x}{1+\cos x} \right) }

Question 10. { \displaystyle {{\tan }^{-1}}\left( \frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right) }

Question 11. { \displaystyle {{\tan }^{-1}}\left( \frac{x}{a+\sqrt{{{a}^{2}}-{{x}^{2}}}} \right) }

Question 12. { \displaystyle {{\sin }^{-1}}\left( \frac{5}{13}\cos x+\frac{12}{13}\sin x \right) }

Question 13. { \displaystyle {{\sin }^{-1}}(x\sqrt{1-x}-\sqrt{x}\sqrt{1-{{x}^{2}}}) }

Questions Discussed:

Simplify:

Question 14. { \displaystyle {{\sin }^{-1}}\left\{ \frac{\sqrt{1+x}+\sqrt{1-x}}{2} \right\} }

Question 15. { \displaystyle {{\tan }^{-1}}\sqrt{\frac{a-x}{a+x}} }

Question 16. { \displaystyle {{\tan }^{-1}}\left( \frac{\cos x}{1-\sin x} \right)},\,\,{ \displaystyle -\frac{\pi }{2}< x < \frac{\pi }{2} }

Question 17. { \displaystyle {{\sin }^{-1}}\left\{ \frac{x+\sqrt{1-{{x}^{2}}}}{\sqrt{2}} \right\} }

Question 18. { \displaystyle {{\tan }^{-1}}(x+\sqrt{1+{{x}^{2}}}) }

Question 19. { \displaystyle \sin \left\{ 2{{\tan }^{-1}}\sqrt{\frac{1-x}{1+x}} \right\} }

Question 20. { \displaystyle {{\sin }^{-1}}\left( \frac{\sin x+\cos x}{\sqrt{2}} \right) }

Question 21. \tan^{-1} \left[ { \displaystyle \frac{a\cos x-b\sin x}{b\cos x+a\sin x} } \right] , if { \displaystyle \frac{a}{b}}\tan x > -1

Question 22. { \displaystyle {{\cot }^{-1}}\left( \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right)}={ \displaystyle\frac{x}{2}},\,\,x\in { \displaystyle \left( 0,\,\,\frac{\pi }{4} \right) }

Question 23. \tan^{-1} \left( { \displaystyle \frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}} } \right ) = { \displaystyle \frac{\pi }{4}-\frac{1}{2}{{\cos }^{-1}}x},\,\,{ \displaystyle-\frac{1}{\sqrt{2}}\le x\le 1 }