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Mathematics LibreTexts

6.3E: Inverse Trigonometric Functions (Exercises)

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    Section 6.3 Exercises

    Evaluate the following expressions, giving the answer in radians.

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

    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. 屏幕快照 2019-07-09 上午4.34.06.png18.屏幕快照 2019-07-09 上午4.34.22.png

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

    Answer

    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. \(\dfrac{3x}{\sqrt{9x^2 + 1}}\)