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Inverse Trig Functions — Practice Problems

Test your ability to evaluate inverse trigonometric functions. Problems range from standard unit-circle values to compositions of trig and inverse trig functions. Click 'Show Answer' for the full solution.

Quick Recap

sin⁡−1(x)\sin^{-1}(x) returns the angle in [−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}] whose sine is xx. cos⁡−1(x)\cos^{-1}(x) returns the angle in [0,π][0, \pi] whose cosine is xx. tan⁡−1(x)\tan^{-1}(x) returns the angle in (−π2,π2)(-\frac{\pi}{2}, \frac{\pi}{2}) whose tangent is xx. These restricted ranges ensure the inverse functions are well-defined.

Practice Problems

1
Evaluate sin⁡−1 ⁣(12)\sin^{-1}\!\left(\frac{1}{2}\right).
2
Evaluate cos⁡−1 ⁣(−22)\cos^{-1}\!\left(-\frac{\sqrt{2}}{2}\right).
3
Evaluate tan⁡−1(1)\tan^{-1}(1).
4
Evaluate sin⁡−1(−1)\sin^{-1}(-1).
5
Evaluate cos⁡−1(0)\cos^{-1}(0).
6
Evaluate tan⁡−1(−3)\tan^{-1}(-\sqrt{3}).
7
Evaluate sin⁡(cos⁡−1(35))\sin(\cos^{-1}(\frac{3}{5})).
8
Evaluate cos⁡(tan⁡−1(512))\cos(\tan^{-1}(\frac{5}{12})).
9
True or false: sin⁡−1(sin⁡(5π6))=5π6\sin^{-1}(\sin(\frac{5\pi}{6})) = \frac{5\pi}{6}.
10
Evaluate tan⁡−1(0)+cos⁡−1(−1)+sin⁡−1(0)\tan^{-1}(0) + \cos^{-1}(-1) + \sin^{-1}(0).

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