8.3E: Inverse Trigonometric Functions (Exercises)

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For the following exercises, find the exact value without the aid of a calculator.

28. \(\sin ^{-1}(1)\)

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

30. \(\tan ^{-1}(-1)\)

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

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

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

34. \(\cos ^{-1}\left(\tan \left(\frac{3 \pi}{4}\right)\right)\)

35. \(\sin \left(\sec ^{-1}\left(\frac{5}{3}\right)\right)\)

36. \(\cot \left(\sin ^{-1}\left(\frac{3}{5}\right)\right)\)

37. \(\tan \left(\cos ^{-1}\left(\frac{5}{13}\right)\right)\)

38. \(\sin \left(\cos ^{-1}\left(\frac{x}{x+1}\right)\right)\)

39. Graph \(f(x)=\cos x\) and \(f(x)=\sec x\) on the interval \([0,2 \pi)\) and explain any observations.

40. Graph \(f(x)=\sin x\) and \(f(x)=\csc x\) and explain any observations.

41. Graph the function \(f(x)=\frac{x}{1}-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}\) on the interval [-1,1] and compare the graph to the graph of \(f(x)=\sin x\) on the same interval. Describe any observations.