## Exercises - Inverse Trigonometric Functions

1. Find the values of the following:

1. $\arcsin 0$

2. $\textrm{arccot}\, (-\frac{\sqrt{3}}{3})$

3. $\cot(\textrm{arccot}\,(-3))$

4. $\cos(\arccos \frac{4}{5})$

5. $\textrm{arccsc}\, 2$

6. $\arccos (-1)$

7. $\csc(\arcsin \frac{3}{5})$

8. $\textrm{arcsec}\, (\sin \frac{\pi}{2})$

9. $\sin (\textrm{arcsec}\, 2)$

10. $\arccos (-\frac{1}{2})$

11. $\arctan^3 (-\sqrt{3})$

12. $3\arcsin^2 (\frac{\sqrt{3}}{2})$

13. $\textrm{arcsec}\, 0$

14. $\sin(\arctan 2)$

15. $\arccos(\sin(-\frac{\pi}{6}))$

16. $\tan(\arccos(-\frac{2}{3}))$

17. $\arccos 2$

18. $\cos(\arcsin(-\frac{4}{5}))$

19. $4 \arctan 1$

20. $\csc(\textrm{arcsec}\, 12)$

21. $\textrm{arccsc}\, \sqrt{2}$

22. $\textrm{arcsec}\, 2$

23. $\arctan(\sin \frac{\pi}{2})$

24. $\arctan(\cos \pi)$

25. $\arcsin (\tan \frac{\pi}{4})$

1. $0$

2. $\frac{2\pi}{3}$

3. $-3$

4. $\frac{4}{5}$

5. $\frac{\pi}{6}$

6. $\pi$

7. $\frac{5}{3}$

8. $0$

9. $\frac{\sqrt{3}}{2}$

10. $\frac{2\pi}{3}$

11. $-\frac{\pi^3}{27}$

12. $\frac{\pi^2}{3}$

13. no value

14. $\frac{2}{\sqrt{5}}$

15. $\frac{2\pi}{3}$

16. $-\frac{\sqrt{5}}{2}$

17. no value

18. $\frac{3}{5}$

19. $\pi$

20. $\frac{12}{\sqrt{143}}$

21. $\frac{\pi}{4}$

22. $\frac{\pi}{3}$

23. $\frac{\pi}{4}$

24. $-\frac{\pi}{4}$

25. $\frac{\pi}{2}$

2. Write the given expression in terms of $x$ without any trigonometric functions:

1. $\sin(\arctan x)$

2. $\tan(\arcsin x)$

3. $\cot(\arcsin x)$

4. $\cos(\arcsin x)$

5. $\cos(\textrm{arcsec}\, x)$

6. $\csc(\textrm{arccot}\, \frac{x}{4})$

1. $\displaystyle{\frac{x}{\sqrt{x^2 + 1}}}$

2. $\displaystyle{\frac{x}{\sqrt{1-x^2}}}$

3. $\displaystyle{\frac{\sqrt{1-x^2}}{x}}$

4. $\displaystyle{\sqrt{1-x^2}}$

5. $\displaystyle{\frac{1}{x}}$

6. $\displaystyle{\frac{\sqrt{x^2 + 16}}{4}}$

3. Write the given expression in terms of $x$ without any trigonometric functions:

1. $\sin(\arctan x)$

2. $\tan(\arcsin x)$

3. $\cot(\arcsin x)$

4. $\cos(\arcsin x)$

5. $\cos(\textrm{arcsec}\, x)$

6. $\csc(\textrm{arccot}\, \frac{x}{4})$

1. $\displaystyle{\frac{x}{\sqrt{x^2 + 1}}}$

2. $\displaystyle{\frac{x}{\sqrt{1-x^2}}}$

3. $\displaystyle{\frac{\sqrt{1-x^2}}{x}}$

4. $\displaystyle{\sqrt{1-x^2}}$

5. $\displaystyle{\frac{1}{x}}$

6. $\displaystyle{\frac{\sqrt{x^2 + 16}}{4}}$