Definition:Inverse Cosine/Real

From ProofWiki
Jump to: navigation, search

Definition

Let $x \in \R$ be a real number such that $-1 \le x \le 1$.

The inverse cosine of $x$ is the multifunction defined as:

$\cos^{-1} \left({x}\right) := \left\{{y \in \R: \cos \left({y}\right) = x}\right\}$

where $\cos \left({y}\right)$ is the cosine of $y$.


Arccosine

Arccosine Function

From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\left[{0 \,.\,.\, \pi}\right]$.

From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.


Therefore, let $g: \left[{0 \,.\,.\, \pi}\right] \to \left[{-1 \,.\,.\, 1}\right]$ be the restriction of $\cos x$ to $\left[{0 \,.\,.\, \pi}\right]$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\left[{-1 \,.\,.\, 1}\right]$.


This function is called arccosine of $x$ and is written $\arccos x$.


Thus:

  • The domain of $\arccos x$ is $\left[{-1 \,.\,.\, 1}\right]$
  • The image of $\arccos x$ is $\left[{0 \,.\,.\, \pi}\right]$.