Calculate the inverse sine (arcsin) of a number using this calculator. For real-number results, the input must be between −1 and 1 (inclusive). Select radians or degrees for the output units.

Arcsin(x) Calculator

Calculate Arcsin of a number

For real results, x must be between -1 and 1 (inclusive).

Arcsin Formula

The inverse sine is defined for real inputs x in the range −1 ≤ x ≤ 1, and it returns the (principal) angle C whose sine is x.

C = \arcsin(x),\quad -\frac{\pi}{2} \le C \le \frac{\pi}{2}
\sin(C) = x

Note: arcsin(x) is the same as sin−1(x) (inverse sine). The range restriction −π/2 to π/2 makes arcsin a single-valued function (the principal value). The arcsin can be used to calculate the angle when the sine of the angle is given.

Arcsin Definition

Arcsin (inverse sine) takes a number x between −1 and 1 and returns the principal angle C (between −π/2 and π/2 radians, or −90° and 90°) such that sin(C) = x.

How to calculate arcsin?

How to calculate arcsin?

  1. First, start with the sine value x.

    Identify the value x (a sine value) and confirm it lies between −1 and 1 for a real-number result.

  2. Next, compute the inverse sine.

    Compute C = arcsin(x) (also written sin⁻¹(x)). This gives the principal angle C in the range −π/2 to π/2.

  3. Finally, convert units if needed.

    If you need degrees, convert from radians: C(deg) = C(rad) × (180/π).

FAQ

What is the difference between arcsin and sin?
For an acute angle in a right-angled triangle, sin refers to the trigonometric function that gives the ratio of the length of the side opposite the angle to the length of the hypotenuse. Arcsin, or inverse sine, is the function that finds the (principal) angle whose sine is a given number (with input between -1 and 1 for real results).

Can arcsin values be in both radians and degrees?
Yes, the arcsin values can be expressed in both radians and degrees. The choice between radians and degrees depends on the context of the problem or the preference of the user.

Is there a specific range for arcsin values?
Yes, the range of arcsin values is restricted to -π/2 to π/2 radians or -90 to 90 degrees because it is the range in which each sine value is unique.

How do you calculate arcsin without a calculator?
Calculating arcsin without a calculator involves understanding the sine values for specific angles and using trigonometric identities or tables. For angles commonly used in trigonometry (like 30°, 45°, 60°), their sine values are well-known, and their corresponding arcsin can be directly stated. For other values, one might need to refer to trigonometric tables or use series expansions.