6.7E: Inverse Trigonometric Functions (Exercises)
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Section 6.3 Exercises
Evaluate the following expressions, giving the answer in radians.
1. \(\sin ^{-1} \left(\dfrac{\sqrt{2} }{2} \right)\)
2. \(\sin ^{-1} \left(\dfrac{\sqrt{3} }{2} \right)\)
3. \(\sin ^{-1} \left(-\dfrac{1}{2} \right)\)
4. \(\sin ^{-1} \left(-\dfrac{\sqrt{2} }{2} \right)\)
5. \(\cos ^{-1} \left(\dfrac{1}{2} \right)\)
6. \(\cos ^{-1} \left(\dfrac{\sqrt{2} }{2} \right)\)
7. \(\cos ^{-1} \left(-\dfrac{\sqrt{2} }{2} \right)\)
8. \(\cos ^{-1} \left(-\dfrac{\sqrt{3} }{2} \right)\)
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. \(\sin ^{-1} \left(\cos \left(\dfrac{\pi }{4} \right)\right)\)
20. \(\cos ^{-1} \left(\sin \left(\dfrac{\pi }{6} \right)\right)\)
21. \(\sin ^{-1} \left(\cos \left(\dfrac{4\pi }{3} \right)\right)\)
22. \(\cos ^{-1} \left(\sin \left(\dfrac{5\pi }{4} \right)\right)\)
23. \(\cos \left(\sin ^{-1} \left(\dfrac{3}{7} \right)\right)\)
24. \(\sin \left(\cos ^{-1} \left(\dfrac{4}{9} \right)\right)\)
25. \(\cos \left(\tan ^{-1} \left(4\right)\right)\)
26. \(\tan \left(\sin ^{-1} \left(\dfrac{1}{3} \right)\right)\)
Find a simplified expression for each of the following.
27. \(\sin \left(\cos ^{-1} \left(\dfrac{x}{5} \right)\right)\), for \(-5\le x\le 5\)
28. \(\tan \left(\cos ^{-1} \left(\dfrac{x}{2} \right)\right)\), for \(-2\le x\le 2\)
29. \(\sin \left(\tan ^{-1} \left(3x\right)\right)\)
30. \(\cos \left(\tan ^{-1} \left(4x\right)\right)\)
- Answer
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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. \(\dfrac{3x}{\sqrt{9x^2 + 1}}\)