The Arccosine Function

Definition The restricted cosine branch is the cosine function with its domain cut down to the closed interval $[0, \pi]$. On this interval cosine decreases from $\cos 0 = 1$ to $\cos \pi = -1$ and takes every value in $[-1, 1]$ exactly once, so it is one-to-one with range $[-1, 1]$. The interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ used for sine would not work, since cosine is even and $\cos\left(-\frac{\pi}{3}\right) = \cos \frac{\pi}{3}$.

Definition For $-1 \le u \le 1$, the arccosine of $u$ is the unique angle $\theta$ in radians with

It is written $\theta = \arccos u$ or $\theta = \cos^{-1} u$, where $\cos^{-1}$ means the inverse function and not the reciprocal $\frac{1}{\cos u} = \sec u$. The inverse identities are

Definition The value $\arccos u$ is the principal angle with cosine $u$. It always lies in $[0, \pi]$, so it is never negative. It is acute when $u > 0$, a right angle when $u = 0$ and obtuse when $u < 0$. For negative inputs the identity

holds, because $\cos(\pi - \theta) = -\cos \theta$ and $\pi - \theta$ stays in $[0, \pi]$ when $\theta$ does. Every other angle with cosine $u$ is $\pm \arccos u + 2k\pi$ for some integer $k$.

Definition Arccosine is decreasing on $[-1, 1]$, which means that $-1 \le u_1 < u_2 \le 1$ implies $\arccos u_1 > \arccos u_2$. It inherits this from the restricted cosine branch, which is decreasing. Arccosine is neither even nor odd. Its graph is instead symmetric about the point $\left(0, \frac{\pi}{2}\right)$, which is what the identity $\arccos(-u) = \pi - \arccos u$ says.

Definition The domain of arccosine is the closed interval $[-1, 1]$. For a composition $\arccos(g(x))$ the domain is the set of $x$ in the domain of $g$ with $-1 \le g(x) \le 1$.

Definition The range of arccosine is the closed interval $[0, \pi]$, the domain of the restricted cosine branch. Every value in it is attained.

Definition The landmark values of arccosine are

The first and last are attained at the included endpoints of the domain, and $\frac{\pi}{2}$ is an interior value. So the graph begins at the solid point $(-1, \pi)$ and ends at the solid point $(1, 0)$.

Definition The graph of $y = \arccos x$ is the set of points $(b, a)$ for which $(a, b)$ lies on the graph of $y = \cos x$ with $0 \le x \le \pi$. Swapping the coordinates is what is meant by reflecting across the line $y = x$, and the picture looks like a mirror image only when both axes use the same scale. The curve falls from $(-1, \pi)$ through $\left(0, \frac{\pi}{2}\right)$ to $(1, 0)$. The cosine branch has horizontal tangent lines at $\theta = 0$ and $\theta = \pi$, so the arccosine graph has vertical tangent lines at its endpoints $(-1, \pi)$ and $(1, 0)$ and falls more and more steeply as $x$ approaches $-1$ or $1$.