Exercises

Precalculus for Everybody

Chapter 4. Real Functions

Inverse Trigonometric Functions

In the exercises, from 1 to 6, evaluate the given expressions without using a calculator.

  1. \[ \sin^{-1} \left( \sqrt{3}/2 \right) \]
  2. \[ \sec^{-1} \left( -\sqrt{2} \right) \]
  3. \[ \cos^{-1}(-1) \]
  4. \[ \tan^{-1} \left( -\sqrt{3} \right) \]
  5. \[ \cot^{-1}(-1) \]
  6. \[ \text{cosec}^{-1}(-2) \]

  7. Given \(y=\sin^{-1} \left( \frac{1}{3} \right)\), find the precise value of:

    1. \[ \cos y \]
    2. \[ \tan y \]
    3. \[ \cot y \]
    4. \[ \sec y \]
    5. \[ \text{cosec } y \]

  8. Given \(y=\sec^{-1}\left( \frac{\sqrt{5}}{2} \right)\), find the precise value of:

    1. \[ \sin y \]
    2. \[ \cos y \]
    3. \[ \tan y \]
    4. \[ \cot y \]
    5. \[ \text{cosec } y \]

  9. Given \(y=\tan^{-1}(-3)\), find the precise value of:

    1. \[ \sin y \]
    2. \[ \cos y \]
    3. \[ \cot y \]
    4. \[ \sec y \]
    5. \[ \text{cosec } y \]

In the exercises, from 10 to 13, find the value of the expression.

  1. \[ \cos^{-1} \left( \sqrt{ \frac{3}{2}} \right) \]
  2. \[ \text{cosec} \left( \tan^{-1} (-2) \right) \]
  3. \[ \sin \left( \tan^{-1} \left( - \frac{3}{4} \right) \right) \]
  4. \[ \tan \left( \sin^{-1} \left( - \frac{3}{4} \right) \right) \]

In the exercises 14 and 15 find the value of the expression.

  1. \[ \sin^{-1} \left( \cos \left( -\frac{\pi}{6} \right) \right) \]
  2. \[ \tan^{-1} \left( \tan \left( \frac{4 \pi}{3} \right) \right) \]

In the exercises, from 16 to 19, find the value of the expression.

  1. \[ \cos \left( \sin^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{3} \right) \right) \]
  2. \[ \sin \left( 2\cos^{-1} \left( \frac{1}{3} \right) \right) \]
  3. \[ \tan \left( 2 \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \right) \]
  4. \[ \cos \left( \left( \frac{1}{2} \right) \sin^{-1} \left( \frac{5}{13} \right) \right) \]

In the exercises, from 20 to 23, find the algebraic expression.

  1. \[ \sin \left( \tan^{-1}(x) \right) \]
  2. \[ \tan \left( \sin^{-1}(x) \right) \]
  3. \[ \sin \left( \cos^{-1} \left( \frac{x}{2} \right) \right) \]
  4. \[ \cos \left( \left( \frac{1}{2} \right) \cos^{-1}(x) \right) \]

Solve the following equations:

  1. \[ \sin^{-1} \left( \frac{x}{2} \right) = -\frac{1}{2} \]
  2. \[ \sin^{-1} \left( \sqrt{2x} \right) = \cos^{-1} x \]
  3. \[ \tan^{2} x + 9 \tan x - 12 = 0 \, , \quad -\frac{\pi}{2} < x < \frac{\pi}{2} \]
    \[ \begin{aligned} &\tan^{2} x + 9 \tan x - 12 = 0 \, , \\[1em] &\hspace{6em} -\frac{\pi}{2} < x < \frac{\pi}{2} \end{aligned} \]
  1. \[ \frac{\pi}{3} \]
  2. \[ \frac{5}{4} \pi \]
  3. \[ \pi \]
  4. \[ -\frac{\pi}{3} \]
  5. \[ \frac{3 \pi}{4} \]
  6. \[ \frac{7\pi}{6} \]
  7.  
    1. \[ \frac{2}{3} \sqrt{2} \]
    2. \[ \frac{1}{4} \sqrt{2} \]
    3. \[ 2 \sqrt{2} \]
    4. \[ \frac{3}{4} \sqrt{2} \]
    5. \[ 3 \]
  8.  
    1. \[ \frac{1}{5} \sqrt{5} \]
    2. \[ \frac{2}{5} \sqrt{5} \]
    3. \[ \frac{1}{2} \]
    4. \[ 2 \]
    5. \[ \sqrt{5} \]
  9.  
    1. \[ -\frac{3 \sqrt{10} }{10} \]
    2. \[ \frac{1}{10}\sqrt{10} \]
    3. \[ -\frac{1}{3} \]
    4. \[ \sqrt{10} \]
    5. \[ -\frac{\sqrt{10}}{3} \]
  10. \[ \frac{1}{2} \]
  11. \[ -\frac{\sqrt{5}}{2} \]
  12. \[ -\frac{3}{5} \]
  13. \[ -\frac{3\sqrt{7}}{7} \]
  14. \[ \frac{\pi}{3} \]
  15. \[ \frac{\pi}{3} \]
  16. \[ \frac{ 2 \sqrt{5} }{5} – \frac{ \sqrt{10} }{30} \]
  17. \[ \frac{4}{9} \sqrt{2} \]
  18. \[ \sqrt{3} \]
  19. \[ \frac{5 \sqrt{26}}{26} \]
  20. \[ \frac{ x }{ \sqrt{ 1 + x^2 } } \]
  21. \[ \frac{x}{ \sqrt{ 1 – x^2 } } \]
  22. \[ \frac{ \sqrt{ 4 – x^2 } }{2} \]
  23. \[ \sqrt{ \frac{ 1 + x }{2} } \]
  24. \[ x = 2 \sin (-0.5) \approx – 0.958851077 \]
  25. \[ x = \sqrt{2} – 1 \]

  26. \(x \approx 0.8673\)   ó   \(x \approx -1.4728682\)