6.3E: Inverse Trigonometric Functions (Exercises)

Section 6.3 Exercises

Evaluate the following expressions, giving the answer in radians. \[1. \sin ^{-1} \left(\frac{\sqrt{2} }{2} \right) 2. \sin ^{-1} \left(\frac{\sqrt{3} }{2} \right) 3. \sin ^{-1} \left(-\frac{1}{2} \right)~ 4. \sin ^{-1} \left(-\frac{\sqrt{2} }{2} \right)\] \[5. \cos ^{-1} \left(\frac{1}{2} \right)~ 6. \cos ^{-1} \left(\frac{\sqrt{2} }{2} \right) 7. \cos ^{-1} \left(-\frac{\sqrt{2} }{2} \right) 8. \cos ^{-1} \left(-\frac{\sqrt{3} }{2} \right)\] \[9. \tan ^{-1} \left(1\right)~ 10. \tan ^{-1} \left(\sqrt{3} \right) 11. \tan ^{-1} \left(-\sqrt{3} \right) 12. \tan ^{-1} \left(-1\right)\]

Use your calculator to evaluate each expression, giving the answer in radians. \[13. \cos ^{-1} \left(-0.4\right) 14. \cos ^{-1} \left(0.8\right) 15. \sin ^{-1} \left(-0.8\right) 16. \tan ^{-1} \left(6\right)\]

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. \(\cos \left(\tan ^{-1} \left(4x\right)\right)\) Section 6.4 Solving Trig Equations 439