Arcsine
The arcsine (notation: arcsin or sin^{-1}) is the inverse function of the sine. That is:
As the arcsine and sine are inverse functions, their composition is the identity. That is:
Properties of Arcsine
- Domain (x): The domain for arcsin x is from −1 to 1,
- Range (θ): The range, or output for arcsin x is all angles from −π/2 to π/2 radians.
To define the inverse function of a function, it must necessarily be bijective. The sine function is not injective in the set of reals. By convention, the codomain is restricted to the interval to make the sine function bijective.
That is, since none of the six trigonometric functions are one-to-one, they must be restricted to have inverse functions. Thus the domains of the trigonometric functions are restricted appropriately so that they become one-to-one functions and their inverse can be determined.
- Symmetry: since sin (-x) = -sin (x) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0, 0).
- Increasing-decreasing behaviour: incresing.
- Continuity: continuous for all x in domain.
- The derivative of arcsine function:
- The integral of arcsine function:
- End behaviour: The limits as x approaches ±∞ do not exist.
Arcsine Table
The arcsine of the most common values is:
Graphical Representation of the Arcsine Function
To better understand the graph of the arcsine, let’s first see the graphical representation of the sine function:
As we see in the graph above the sine is periodic, it is not one-to-one and the graph of the sine function fails the horizontal line test. Hence the sine does not have an inverse unless we restrict its domain. So, by convention, the domain of the sine is usually restricted to the interval (-π/2, π/2).
The graph of the arcsine function is symmetric to that of the sine function to the bisector line of the first and third quadrants (y = x). With the restriction on the interval (-π/2, π/2), as seen above, both functions are increasing and one is inverse of the other.