Definition:Inverse Cosine/Real/Arccosine

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Definition

Real Arccosine Function

From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$.

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


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

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$.


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


Thus:

The domain of $\arccos x$ is $\closedint {-1} 1$
The image of $\arccos x$ is $\closedint 0 \pi$.


Terminology

There exists the popular but misleading notation $\cos^{-1} x$, which is supposed to denote the inverse cosine function.

However, note that as $\cos x$ is not an injection (even though by restriction of the codomain it can be considered surjective), it does not have a well-defined inverse.

The $\arccos$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\cos^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname{Cos}^{-1}$ (with a capital $\text C$) is taken to mean the same as $\arccos$.

However, this can also be confusing due to the visual similarity between that and the lower case $\text c$.


Some sources hyphenate: arc-cosine.


Also denoted as

The symbol used to denote the arccosine function is variously seen as:

  • $\arccos$
  • $\operatorname {acos}$


Examples

Example: $\map \sin {2 \arccos x}$

$\map \sin {2 \arccos x}$

can be simplified to:

$2 x \sqrt {1 - x^2}$


Also see

  • Results about inverse cosine can be found here.


Other inverse trigonometrical ratios


Sources

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