Inverse Sine Calculator

What is an inverse sine calculator?

The inverse sine calculator finds the angle whose sine equals a value you provide. The sine function takes an angle and returns a ratio between -1 and 1; the inverse sine (also called arcsine) runs that relationship backwards, taking a ratio and returning the angle that produced it. Enter any sine value in the range from -1 to 1 and the calculator reports the resulting angle in both degrees and radians.

Because the sine function repeats and is not one-to-one over all angles, the arcsine is defined on a restricted range. This calculator returns the principal value: an angle between $-90^\circ$ and $90^\circ$ (equivalently between $-\tfrac{\pi}{2}$ and $\tfrac{\pi}{2}$ radians).

How does it work?

The arcsine is written $\arcsin$ or $\sin^{-1}$. For a sine value $x$ the angle $\theta$ is

$\theta = \arcsin(x)$

The calculator evaluates the arcsine in radians and then converts to degrees using

$\theta_{\deg} = \arcsin(x) \cdot \frac{180}{\pi}$

If you enter a value outside the interval $[-1, 1]$, no real angle has that sine, so the calculator leaves the result blank.

Worked examples

• A sine value of $0.5$ gives $\arcsin(0.5) = 30^\circ$, which is $0.5236$ radians.
• A sine value of $0.7071$ gives $\arcsin(0.7071) = 45^\circ$, which is $\tfrac{\pi}{4} \approx 0.7854$ radians.
• A sine value of $1$ gives $\arcsin(1) = 90^\circ$, the largest angle the function returns.
• A sine value of $0$ gives $\arcsin(0) = 0^\circ$.

Practical notes

The inverse sine is essential whenever you know a ratio of sides in a right triangle and need the angle. For example, if the side opposite an angle is half the hypotenuse, the ratio is $0.5$ and the angle is $30^\circ$. It also appears in physics for problems involving wave amplitude, projectile launch angles, and refraction.

Keep in mind that the principal value returned here is only one of infinitely many angles with the same sine. For an angle in the second quadrant, for instance, you can use the identity $\sin(\theta) = \sin(180^\circ - \theta)$ to recover the alternative solution. To go the other direction and start from an angle, use the trigonometry calculator. For the related inverse functions, see the inverse cosine and inverse tangent calculators.