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. \(x \approx 0.8673\)   or   \(x \approx -1.4728682\)

In exercises 1 through 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 exercises 10 through 13, find the value of the expression.

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

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

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

In exercises 16 through 19, find the value of the expression.

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

In exercises 20 through 23, find the algebraic expression.

20. \[ \sin \left( \tan^{-1}(x) \right) \]
21. \[ \tan \left( \sin^{-1}(x) \right) \]
22. \[ \sin \left( \cos^{-1} \left( \frac{x}{2} \right) \right) \]
23. \[ \cos \left( \left( \frac{1}{2} \right) \cos^{-1}(x) \right) \]

Solve the following equations:

24. \[ \sin^{-1} \left( \frac{x}{2} \right) = -\frac{1}{2} \]
25. \[ \sin^{-1} \left( \sqrt{2x} \right) = \cos^{-1} x \]
26. \[ \tan^{2} x + 9 \tan x – 12 = 0 \, , \quad -\frac{\pi}{2} < x < \frac{\pi}{2} \]
    \[ \begin{aligned} &\tan^{2} x + 9 \tan x – 12 = 0 \, , \\[1em] &\hspace{6em} -\frac{\pi}{2} < x < \frac{\pi}{2} \end{aligned} \]