# Arcsine and Arccosine
Source: algebrica.org — CC BY-NC 4.0
https://algebrica.org/arcsine-and-arccosine/
## Arcsine
The arcsine is the inverse of the [sine](../sine-and-cosine.md) function. Given a number \\(x \in [-1, 1]\\) (i.e., the range of values the sine function can attain), \\(\arcsin(x)\\) is defined as the angle \\(\theta\\) in the interval \\([-\pi/2, \pi/2]\\) whose sine is equal to \\(x\\). In general, an [inverse function](../inverse-function.md) reverses the operation of the original: if a function \\(f\\) maps a value \\(x\\) to a value \\(y\\), then its inverse \\(f^{-1}\\) maps \\(y\\) back to \\(x\\). The sine function takes an angle and returns a real number in \\([-1, 1]\\) and the arcsine does the opposite, returning the angle whose sine equals the given value. This inverse relationship is expressed by the identity:
\\[
\sin(\arcsin(x)) = x \quad \forall \\, x \in [-1, 1]
\\]
In formal terms, the definition of the arcsine is the following:
\\[
\arcsin(x) = \theta \quad \iff \quad \sin(\theta) = x \quad \text{and} \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
\\]
The restriction of \\(\theta\\) to the interval \\(\left[-\pi/2, \pi/2 \right]\\) is necessary because the sine function is not injective on its full [domain](../determining-the-domain-of-a-function.md). Without this restriction, the inverse would not be well-defined.
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## Example
Consider the computation of \\(\arcsin\\!\left(\frac{1}{2}\right)\\). We seek the angle \\(\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\\) such that \\(\sin(\theta) = \frac{1}{2}\\). From the standard values of the sine function, we know that:
\\[
\sin\\!\left(\frac{\pi}{6}\right) = \frac{1}{2}
\\]
Since \\(\frac{\pi}{6}\\) belongs to the interval \\(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\\), it satisfies all the conditions required by the definition. We conclude that:
\\[\arcsin\\!\left(\frac{1}{2}\right) = \frac{\pi}{6}\\]
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## Common values of the arcsine
The following table collects the standard values of \\(\arcsin(x)\\) for the most frequently encountered inputs:
\\[
\begin{align}
x &= -1 &\quad& \arcsin(-1) = -\pi/2 \\\\[6pt]
x &= -\sqrt{3}/2 &\quad& \arcsin(-\sqrt{3}/2) = -\pi/3 \\\\[6pt]
x &= -1/2 &\quad& \arcsin(-1/2) = -\pi/6 \\\\[6pt]
x &= 0 &\quad& \arcsin(0) = 0 \\\\[6pt]
x &= 1/2 &\quad& \arcsin(1/2) = \pi/6 \\\\[6pt]
x &= \sqrt{3}/2 &\quad& \arcsin(\sqrt{3}/2) = \pi/3 \\\\[6pt]
x &= 1 &\quad& \arcsin(1) = \pi/2
\end{align}
\\]
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## Arccosine
The arccosine is the inverse of the [cosine](../sine-and-cosine.md) function. Given a number \\(x \in [-1, 1]\\) (i.e., the range of values the cosine function can attain), \\(\arccos(x)\\) is defined as the angle \\(\theta\\) in the interval \\([0, \pi]\\) whose cosine is equal to \\(x\\). As with the arcsine, the restriction of the codomain to \\([0, \pi]\\) is necessary to ensure that the inverse is well-defined, since the cosine function is not injective on its full domain. The corresponding identity is:
\\[
\cos(\arccos(x)) = x \quad \text{for all } x \in [-1, 1]
\\]
In formal terms, the definition of the arccosine is the following:
\\[
\arccos(x) = \theta \quad \text{if and only if} \quad \cos(\theta) = x \quad \text{and} \quad \theta \in [0, \pi]
\\]
- - -
## Common values of the arccosine
The following table collects the standard values of \\(\arccos(x)\\) for the most frequently encountered inputs:
\\[
\begin{align}
x &= -1 &\quad& \arccos(-1) = \pi \\\\[6pt]
x &= -\sqrt{3}/2 &\quad& \arccos(-\sqrt{3}/2) = 5\pi/6 \\\\[6pt]
x &= -1/2 &\quad& \arccos(-1/2) = 2\pi/3 \\\\[6pt]
x &= 0 &\quad& \arccos(0) = \pi/2 \\\\[6pt]
x &= 1/2 &\quad& \arccos(1/2) = \pi/3 \\\\[6pt]
x &= \sqrt{3}/2 &\quad& \arccos(\sqrt{3}/2) = \pi/6 \\\\[6pt]
x &= 1 &\quad& \arccos(1) = 0
\end{align}
\\]
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## Properties of the arcsine and arccosine
The arcsine and arccosine functions are related by the following identity, which holds for every \\(x \in [-1, 1]\\):
\\[
\arcsin(x) + \arccos(x) = \frac{\pi}{2}
\\]
This identity reflects the complementary nature of the two functions: since the sine and cosine of complementary angles are equal, the angle whose sine is \\(x\\) and the angle whose cosine is \\(x\\) always sum to \\(\pi/2\\).
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A second property worth noting concerns the composition of a function with its inverse. One direction is straightforward: applying the arcsine after the sine, or the arccosine after the cosine, recovers the original value, provided the argument lies in the appropriate interval. Formally:
\\[
\sin(\arcsin(x)) = x \quad \forall x \in [-1, 1]
\\]
\\[
\cos(\arccos(x)) = x \quad \forall x \in [-1, 1]
\\]
The opposite composition, however, does not hold in general. For an arbitrary angle \\(\theta\\), one has:
\\[
\arcsin(\sin(\theta)) = \theta \quad \iff \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
\\]
\\[
\arccos(\cos(\theta)) = \theta \quad \iff \quad \theta \in [0, \pi]
\\]
> Outside these intervals, the arcsine and arccosine return the unique representative of \\(\theta\\) within their respective ranges, not \\(\theta\\) itself. This asymmetry is a direct consequence of the domain restrictions imposed to make the inverses well-defined, and it is what distinguishes a true inverse from a mere left or right inverse.
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## Arcsine and arccosine functions
The arcsine function \\(f(x) = \arcsin(x)\\) assigns to each value \\(x \in [-1, 1]\\) the angle \\(\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\\) whose sine equals \\(x\\). Its graph is a continuous, strictly increasing curve.
* [Domain](../determining-the-domain-of-a-function.md): \\(x \in [-1, 1]\\)
* Range: \\(y \in [-\pi/2, \pi/2]\\)
* Periodicity: the arcsine function is not periodic.
* Parity: the function is [odd](../even-and-odd-functions.md), satisfying \\(\arcsin(-x) = -\arcsin(x)\\).
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The arccosine function \\(f(x) = \arccos(x)\\) assigns to each value \\(x \in [-1, 1]\\) the angle \\(\theta \in [0, \pi]\\) whose cosine equals \\(x\\). Its graph is a continuous, strictly decreasing curve.
* Domain: \\(x \in [-1, 1]\\)
* Range: \\(y \in [0, \pi]\\)
* Periodicity: the arccosine function is not periodic.
* Parity: the function is neither odd nor even, but satisfies the identity \\(\arccos(-x) = \pi - \arccos(x)\\).
