6.3E: Inverse Trigonometric Functions (Exercises)

Last updated
Save as PDF

Page ID: 13925

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

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. $屏幕快照 2019-07-09 上午4.34.06.png$ 18. $屏幕快照 2019-07-09 上午4.34.22.png$

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

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. $\dfrac{3x}{\sqrt{9x^2 + 1}}$