Inverse Sine Function Calculator in Degrees

The inverse sine function, also known as arcsine, calculates the angle whose sine is a given value. This calculator computes the inverse sine in degrees, providing both the principal value and all possible values within the domain of the function.

What is the Inverse Sine Function?

The inverse sine function, written as arcsin(x) or sin⁻¹(x), is the inverse operation of the sine function. While the sine function takes an angle and returns a ratio, the inverse sine function takes a ratio and returns an angle.

The sine function is periodic and not one-to-one over its entire domain, which means it doesn't have a true inverse. However, by restricting the domain to the interval [-π/2, π/2] radians (or [-90°, 90°] degrees), we can define a one-to-one function that has an inverse.

The range of the inverse sine function is [-90°, 90°] when working in degrees. This means the output will always be an angle between -90 and 90 degrees.

How to Use This Calculator

Enter a value between -1 and 1 in the input field. This represents the sine of the angle you want to find.
Click the "Calculate" button to compute the inverse sine in degrees.
The calculator will display the principal value (the angle between -90° and 90°) and all possible values within the domain of the sine function.
Review the explanation of the result and the formula used for the calculation.

Formula and Calculation

The inverse sine function in degrees is calculated using the following formula:

arcsin(x) = sin⁻¹(x) = y (in degrees)

Where:

x is the input value (must be between -1 and 1)
y is the output angle in degrees

The calculator uses the JavaScript Math.asin() function, which returns the arcsine in radians, and then converts it to degrees by multiplying by 180/π.

Worked Examples

Example 1: Finding the Angle

If sin(θ) = 0.5, what is θ in degrees?

Using the inverse sine function:

θ = arcsin(0.5) = 30°

This means the angle whose sine is 0.5 is 30 degrees.

Example 2: Negative Value

If sin(θ) = -0.866, what is θ in degrees?

Using the inverse sine function:

θ = arcsin(-0.866) ≈ -60°

This means the angle whose sine is -0.866 is approximately -60 degrees.

Applications of Inverse Sine

The inverse sine function has several practical applications in mathematics, physics, and engineering:

Solving right triangles when you know the length of the opposite side and the hypotenuse.
Calculating angles in wave motion and signal processing.
Determining the angle of elevation or depression in projectile motion.
Analyzing circular motion and rotational dynamics.

Frequently Asked Questions

What is the domain of the inverse sine function?

The domain of the inverse sine function is all real numbers between -1 and 1, inclusive. This means the input value must be in the range [-1, 1].

What is the range of the inverse sine function in degrees?

The range of the inverse sine function in degrees is [-90°, 90°]. This means the output will always be an angle between -90 and 90 degrees.

Can the inverse sine function return multiple values?

While the principal value of the inverse sine function is unique, there are infinitely many angles that have the same sine value due to the periodic nature of the sine function. However, the calculator shows the principal value within the restricted domain.

How is the inverse sine function different from the sine function?

The sine function takes an angle as input and returns a ratio, while the inverse sine function takes a ratio as input and returns an angle. They are essentially inverse operations of each other.