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# 6.3E: Inverse Trigonometric Functions (Exercises)

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Section 6.3 Exercises

1. $$\sin ^{-1} \left(\frac{\sqrt{2} }{2} \right)$$

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

3. $$\sin ^{-1} \left(-\frac{1}{2} \right)$$

4. $$\sin ^{-1} \left(-\frac{\sqrt{2} }{2} \right)$$

5. $$\cos ^{-1} \left(\frac{1}{2} \right)$$

6. $$\cos ^{-1} \left(\frac{\sqrt{2} }{2} \right)$$

7. $$\cos ^{-1} \left(-\frac{\sqrt{2} }{2} \right)$$

8. $$\cos ^{-1} \left(-\frac{\sqrt{3} }{2} \right)$$

9. $$\tan ^{-1} \left(1\right)$$

10. $$\tan ^{-1} \left(\sqrt{3} \right)$$

11. $$\tan ^{-1} \left(-\sqrt{3} \right)$$

12. $$\tan ^{-1} \left(-1\right)$$

13. $$\cos ^{-1} \left(-0.4\right)$$

14. $$\cos ^{-1} \left(0.8\right)$$

15. $$\sin ^{-1} \left(-0.8\right)$$

16. $$\tan ^{-1} \left(6\right)$$

Find the angle $$\theta$$ in degrees.

17. 18.

Evaluate the following expressions.

19. $$\sin ^{-1} \left(\cos \left(\frac{\pi }{4} \right)\right)$$

20. $$\cos ^{-1} \left(\sin \left(\frac{\pi }{6} \right)\right)$$

21. $$\sin ^{-1} \left(\cos \left(\frac{4\pi }{3} \right)\right)$$

22. $$\cos ^{-1} \left(\sin \left(\frac{5\pi }{4} \right)\right)$$

23. $$\cos \left(\sin ^{-1} \left(\frac{3}{7} \right)\right)$$

24. $$\sin \left(\cos ^{-1} \left(\frac{4}{9} \right)\right)$$

25. $$\cos \left(\tan ^{-1} \left(4\right)\right)$$

26. $$\tan \left(\sin ^{-1} \left(\frac{1}{3} \right)\right)$$

Find a simplified expression for each of the following.

27. $$\sin \left(\cos ^{-1} \left(\frac{x}{5} \right)\right)$$, for $$-5\le x\le 5$$

28. $$\tan \left(\cos ^{-1} \left(\frac{x}{2} \right)\right)$$, for $$-2\le x\le 2$$

29. $$\sin \left(\tan ^{-1} \left(3x\right)\right)$$

30. $$\cos \left(\tan ^{-1} \left(4x\right)\right)$$

1. $$\dfrac{\pi}{4}$$

3. $$-\dfrac{\pi}{6}$$

5. $$\dfrac{\pi}{3}$$

7. $$\dfrac{3\pi}{4}$$

9. $$\dfrac{\pi}{4}$$

11. $$-\dfrac{\pi}{3}$$

13. 1.9823

15. -0.9273

17. $$44.427^{\circ}$$

19. $$\dfrac{\pi}{4}$$

21. $$-\dfrac{\pi}{6}$$

23. $$\dfrac{2\sqrt{10}}{7}$$

25. $$\dfrac{1}{\sqrt{17}}$$

27. $$\dfrac{\sqrt{25-x^2}}{5}$$

29. $$\dfrac{3x}{\sqrt{9x^2 + 1}}$$