Inverse cosine function

The inverse cosine of $ u$, denoted by  $ \cos^{-1} u$, is defined as the unique value $ 0\le\theta\le\pi$ satisfying $ \cos\theta = u$. The inverse cosine function `` $ \cos^{-1} x$'' is also known as the arccosine function ``$ \arccos x$.''

Domain and range of $ \cos^{-1}$. $ D = [-1,\:1]$ and $ R = \left[0,\:\pi\right]$.

\includegraphics{lec14b.ps}

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 $ \theta_1$ and $ \theta_2$ $ ^\dagger$ as

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



Department of Mathematics
Last modified: 2005-09-29