Home fsc 1st year math solutions Solution of inverse trigonometric functions Exercise 13.2

Solution of inverse trigonometric functions Exercise 13.2

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Inverse Trigonometric Functions ex 13.2
Inverse Trigonometric Functions ex 13.2

In this post, we will provide the Solution of inverse trigonometric functions Exercise 13.2. All the questions will be solved step by step and provide you better solutions for best understandings and learning.

Prove the following inverse trigonometric functions:

  1. \sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{7}{25}=\cos ^{-1} \frac{253}{325}
  2. \tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{1}{5}=\tan ^{-1} \frac{9}{19}
  3. 2 \tan ^{-1} \frac{2}{3}=\sin ^{-1} \frac{12}{13} \quad [Hint: Let \tan ^{-1} \frac{2}{3}=x. and \left.\operatorname{show} \operatorname{Sin} 2 x=\frac{12}{13}\right]
  4. \tan ^{-1} \frac{120}{119}=2 \cos ^{-1} \frac{12}{13}
  5. \sin ^{-1} \frac{1}{\sqrt{5}}+\cot ^{-1} 3=\frac{\pi}{4}
  6. \sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{8}{17}=\sin ^{-1} \frac{77}{85}
  7. \sin ^{-1} \frac{77}{85}-\sin ^{-1} \frac{3}{5}=\cos ^{-1} \frac{15}{17}
  8. \cos ^{-1} \frac{63}{65}+2 \tan ^{-1} \frac{1}{5}=\sin ^{-1} \frac{3}{5}
  9. \tan ^{-1} \frac{3}{4}+\tan ^{-1} \frac{3}{5}-\tan ^{-1} \frac{8}{19}=\frac{\pi}{4}
    [Hint: First add \tan ^{-1} \frac{3}{4}+\tan ^{-1} \frac{3}{5} and then proceed]
  10. \sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{16}{65}=\frac{\pi}{2}
  11. \tan ^{-1} \frac{1}{11}+\tan ^{-1} \frac{5}{6}=\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{2}
  12. 2 \tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{7}=\frac{\pi}{4}
  13. Show that \cos \left(\sin ^{-1} x\right)=\sqrt{1-x^2}
  14. Show that \sin \left(2 \cos ^{-1} x\right)=2 x \sqrt{1-x^2}
  15. Show that \cos \left(2 \sin ^{-1} x\right)=1-2 x^2
  16. Show that \tan ^{-1}(-x)=-\tan ^{-1} x
  17. Show that \sin ^{-1}(-x) \quad=-\sin ^{-1} x
  18. Show that \cos ^{-1}(-x)=\pi-\cos ^{-1} x
  19. Show that \tan \left(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^2}}
  20. Given that x=\sin ^{-1} \frac{1}{2}, find the values of following trigonometric functions: \sin x, \cos x, \tan x, \cot x, \sec x and \csc x.

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Inverse trigonometric functions

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