Precalculus Review

Inverse cosine function

The inverse cosine of

, denoted by $\cos^{-1} u$ , is defined as the unique value $0\le\theta\le\pi$ satisfying $\cos\theta = u$ . It is also known as the arccosine function $\arccos x$ . The domain and the range are given by $D = [-1,\:1]$ and $R = \left[0,\:\pi\right]$ .

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Trigonometric equation. Consider the solutions in the cycle $[0,2\pi)$ to the equation ${\cos\theta = u}$ . By observing $\cos(2\pi - \theta) = \cos(-\theta) = \cos\theta$ , we obtain the two solutions

$\displaystyle \theta_1 = \cos^{-1} u$ and $\displaystyle \quad \theta_2 = 2\pi - \cos^{-1} u$