Arcsine and Arccosine
Arcsine
The arcsine is the inverse of the sine 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 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:
\[sin ( arcsin ( x ) ) = x \forall x \in [ - 1 , 1 ]\]
In formal terms, the definition of the arcsine is the following:
\[arcsin ( x ) = \theta \Longleftrightarrow sin ( \theta ) = x \text{and} \theta \in [ - \frac{\pi}{2} , \frac{\pi}{2} ]\]The restriction of $\theta$ to the interval $[ - \pi / 2 , \pi / 2 ]$ is necessary because the sine function is not injective on its full domain. Without this restriction, the inverse would not be well-defined.
Example
Consider the computation of $arcsin ( \frac{1}{2} )$. We seek the angle $\theta \in [ - \frac{\pi}{2} , \frac{\pi}{2} ]$ such that $sin ( \theta ) = \frac{1}{2}$. From the standard values of the sine function, we know that:
\[sin ( \frac{\pi}{6} ) = \frac{1}{2}\]Since $\frac{\pi}{6}$ belongs to the interval $[ - \frac{\pi}{2} , \frac{\pi}{2} ]$, it satisfies all the conditions required by the definition.
We conclude that: \(arcsin ( \frac{1}{2} ) = \frac{\pi}{6}\)
Common values of the arcsine
The following table collects the standard values of $arcsin ( x )$ for the most frequently encountered inputs:
\[x & = - 1 & & arcsin ( - 1 ) = - \pi / 2 \\ x & = - \sqrt{3} / 2 & & arcsin ( - \sqrt{3} / 2 ) = - \pi / 3 \\ x & = - 1 / 2 & & arcsin ( - 1 / 2 ) = - \pi / 6 \\ x & = 0 & & arcsin ( 0 ) = 0 \\ x & = 1 / 2 & & arcsin ( 1 / 2 ) = \pi / 6 \\ x & = \sqrt{3} / 2 & & arcsin ( \sqrt{3} / 2 ) = \pi / 3 \\ x & = 1 & & arcsin ( 1 ) = \pi / 2\]Arccosine
The arccosine is the inverse of the cosine 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:
\[cos ( arccos ( x ) ) = x \text{for all} x \in [ - 1 , 1 ]\]
In formal terms, the definition of the arccosine is the following:
\[arccos ( x ) = \theta \text{if and only if} cos ( \theta ) = x \text{and} \theta \in [ 0 , \pi ]\]Common values of the arccosine
The following table collects the standard values of $arccos ( x )$ for the most frequently encountered inputs:
\[x & = - 1 & & arccos ( - 1 ) = \pi \\ x & = - \sqrt{3} / 2 & & arccos ( - \sqrt{3} / 2 ) = 5 \pi / 6 \\ x & = - 1 / 2 & & arccos ( - 1 / 2 ) = 2 \pi / 3 \\ x & = 0 & & arccos ( 0 ) = \pi / 2 \\ x & = 1 / 2 & & arccos ( 1 / 2 ) = \pi / 3 \\ x & = \sqrt{3} / 2 & & arccos ( \sqrt{3} / 2 ) = \pi / 6 \\ x & = 1 & & arccos ( 1 ) = 0\]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 $\pi / 2$.
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:
\[sin ( arcsin ( x ) ) = x \forall x \in [ - 1 , 1 ]\] \[cos ( arccos ( x ) ) = x \forall x \in [ - 1 , 1 ]\]The opposite composition, however, does not hold in general. For an arbitrary angle $\theta$, one has:
\[arcsin ( sin ( \theta ) ) = \theta \Longleftrightarrow \theta \in [ - \frac{\pi}{2} , \frac{\pi}{2} ]\] \[arccos ( cos ( \theta ) ) = \theta \Longleftrightarrow \theta \in [ 0 , \pi ]\]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.
Arcsine and arccosine functions
The arcsine function $f ( x ) = arcsin ( x )$ assigns to each value $x \in [ - 1 , 1 ]$ the angle $\theta \in [ - \frac{\pi}{2} , \frac{\pi}{2} ]$ whose sine equals $x$. Its graph is a continuous, strictly increasing curve.
- Domain: $x \in [ - 1 , 1 ]$
- Range: $y \in [ - \pi / 2 , \pi / 2 ]$
- Periodicity: the arcsine function is not periodic.
- Parity: the function is odd, satisfying $arcsin ( - x ) = - arcsin ( x )$.
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 $arccos ( - x ) = \pi - arccos ( x )$.
comparisonsfunction behaviorinverse mappingunit circleangle interpretationstandard valuesgraphssymmetry relationsnon periodicityparitycontinuitymonotonicityrangedomainprincipal valuesrange selectiondomain restrictionarccosine definitionarcsine definitionrepresentationspropertiesstructure