Arcsin Calculator - Inverse Sine
Enter any value between -1 and 1 to find the corresponding angle using the inverse sine function (arcsin). The calculator gives the result in both degrees and radians, shows you the step-by-step working, derives the related cosine and tangent of the angle, and can recover a missing angle from two sides of a right triangle.
Formula
Worked example
Right triangle with opposite side a = 3 and hypotenuse c = 5: sin(θ) = 3/5 = 0.6, so θ = arcsin(0.6) ≈ 36.87° (0.6435 rad). The complementary angle is 90° - 36.87° = 53.13°, and cos(θ) = √(1 - 0.36) ≈ 0.8.
What is arcsin (inverse sine)?
The arcsine function, written arcsin(x) or sin^{-1}(x), is the inverse of the sine function. Given a ratio x between -1 and 1, arcsin returns the angle whose sine equals that ratio. Because the sine function repeats every 360 degrees, the inverse is defined only on the principal branch, returning angles in the range -90 degrees to 90 degrees (-pi/2 to pi/2 radians). Outside that range there are infinitely many angles with the same sine, so the principal value is the standard convention agreed on worldwide.
How to use this calculator
Select "Enter sine value" mode and type any number between -1 and 1. The angle appears instantly in both degrees and radians, along with the cosine and tangent of that angle and the complementary angle. Switch to "Right triangle" mode to recover a missing angle from two side lengths: enter the side opposite the angle and the hypotenuse, and the calculator forms the ratio for you before applying arcsin. The "Show your work" panel traces every algebraic step so you can follow the working or copy it for homework.
The Pythagorean identity outputs
Once the angle theta is known, its cosine and tangent follow immediately from the Pythagorean identity sin^2(theta) + cos^2(theta) = 1. Rearranging gives cos(arcsin(x)) = sqrt(1 - x^2) and tan(arcsin(x)) = x / sqrt(1 - x^2). These are displayed as extra outputs so you can use the single-input arcsin step as the foundation for a fuller right-triangle solution without a separate calculator. Note that tan is undefined at x = 1 and x = -1, where the angle is exactly +90 degrees or -90 degrees.
Arcsin and arccos: the complementary relationship
For any x in the domain [-1, 1], arcsin(x) + arccos(x) = 90 degrees (pi/2 radians). This means the complementary angle output equals arccos(x). Knowing this identity is useful when switching between opposite-side and adjacent-side problems in a right triangle: the angle at one vertex is the arcsin of one ratio and the arccos of another, and they always sum to 90 degrees. The derivative of arcsin(x) is 1 / sqrt(1 - x^2), and its integral is x * arcsin(x) + sqrt(1 - x^2) + C, both of which appear in calculus courses.
Common arcsin values
| x (sine value) | arcsin(x) in degrees | arcsin(x) in radians |
|---|---|---|
| -1 | -90° | -π/2 |
| -√3/2 ≈ -0.8660 | -60° | -π/3 |
| -√2/2 ≈ -0.7071 | -45° | -π/4 |
| -1/2 = -0.5 | -30° | -π/6 |
| 0 | 0° | 0 |
| 1/2 = 0.5 | 30° | π/6 |
| √2/2 ≈ 0.7071 | 45° | π/4 |
| √3/2 ≈ 0.8660 | 60° | π/3 |
| 1 | 90° | π/2 |
Exact angles for the standard sine ratios. These arise from the 30-60-90 and 45-45-90 special triangles.
Frequently asked questions
What values can I enter into arcsin?
Only values between -1 and 1 inclusive. The sine of any angle is always in that range, so ratios outside it are impossible in a real right triangle. If you enter a value outside [-1, 1], the result is undefined (in complex numbers a result does exist, but this calculator covers real-valued angles only).
What is the difference between arcsin and sin^{-1}?
They mean exactly the same thing. The notation sin^{-1}(x) is read as "the inverse sine of x" and equals arcsin(x). The superscript -1 here denotes an inverse function, NOT the reciprocal 1/sin(x), which is the cosecant. To avoid confusion, many textbooks prefer the "arcsin" spelling.
Why does arcsin only return values between -90 and 90 degrees?
The sine function is periodic: sin(30 degrees) = sin(150 degrees) = sin(390 degrees), and so on. To make the inverse well-defined (one output per input), mathematicians restrict the domain of sine to -90 degrees to 90 degrees before inverting it. The result is the principal value. All other angles with the same sine are given by theta + 360 degrees * k or (180 degrees - theta) + 360 degrees * k for integer k.
How do I find all angles with a given sine, not just the principal value?
Let theta_0 = arcsin(x) be the principal value. The full set of solutions is theta = theta_0 + 360 degrees * k and theta = 180 degrees - theta_0 + 360 degrees * k, for any integer k. For example, arcsin(0.5) = 30 degrees, but 150 degrees, 390 degrees, and -330 degrees all have sine 0.5 as well.
What is arcsin used for in practice?
Arcsin is used whenever you need to recover an angle from a known sine ratio. Common applications include finding a missing angle in a right triangle (surveying, navigation, construction), converting between polar and Cartesian coordinates, computing phase angles in signal processing, and finding angles of incidence or refraction in optics.