Arcsine Function
What Is the Arcsine Function?
The arcsine function, denoted by arcsin, is the inverse of the sine function when the sine is restricted to the interval [-π/2, π/2]. Its purpose is simple: given a sine value, it returns the angle whose sine equals that value.
You may also encounter the notation sin-1. In this case, the exponent -1 indicates an inverse function, not a reciprocal. Because this notation can sometimes be misunderstood, many textbooks and scientific publications prefer the notation arcsin.
How Does the Arcsine Function Work?
The arcsine function starts with a sine value and works backward to find the corresponding angle.
For example, suppose you know that
$$ \sin x = 0.75 $$
What is the angle x?
To find the answer, compute the arcsine of 0.75:
$$ x = \arcsin(0.75) \approx 0.848 $$
This means that an angle of approximately 0.848 radians has a sine equal to 0.75.
On the unit circle, the same value also represents the length of the corresponding arc, which explains the origin of the name arcsine.
Why Can't the Sine Function Be Inverted Everywhere?
The sine function takes values between -1 and 1.
However, it is also a periodic function. As the angle increases, the same sine values repeat over and over again.
For example, many different angles have a sine value of 0.75. Because of this, the sine function is not one-to-one over its entire domain.
An inverse function can only exist when each output value corresponds to exactly one input value. Since multiple angles can produce the same sine value, the sine function cannot be inverted on all real numbers.
How Is the Arcsine Function Defined?
To create an inverse, we restrict the domain of the sine function to an interval where every sine value occurs only once.
The standard choice is the interval [-π/2, π/2].
Within this interval, each value in the range [-1, 1] is associated with exactly one angle.
As a result, the restricted sine function becomes one-to-one and can be inverted.
This inverse function is called the arcsine function.
Formally, the arcsine function is defined as
$$ \arcsin : [-1,1] \rightarrow [-\pi/2,\pi/2] $$
For every number x between -1 and 1, the value arcsin(x) is the unique angle in the interval [-π/2, π/2] whose sine equals x.
Note: The interval [-π/2, π/2] is known as the principal branch of the arcsine function. Although it is the standard choice, it is not the only possible one. Any interval on which the sine function is one-to-one can be used to define a valid inverse branch.
How to Graph the Arcsine Function
The graph of the arcsine function is closely related to the graph of the sine function.
Start with the graph of the sine function restricted to the interval [-π/2, π/2]. Since the arcsine is the inverse of this restricted function, its graph is obtained by reflecting the sine graph across the line y = x.
This is a general property of inverse functions: the graph of a function and the graph of its inverse are mirror images of each other with respect to the line y = x.
The same idea applies to any other interval on which the sine function is one-to-one. Restrict the sine function to that interval and then reflect the graph across the line y = x to obtain the corresponding inverse branch.
