Inverse Sine Calculator

§1What arcsin means

Sine takes an angle and returns a number: sin 30° = 1/2. Inverse sine goes the other way. It takes the number and returns the angle: arcsin(1/2) = 30°, or π/6 in radians.

On the unit circle, sin θ is the y-coordinate of the point on the circle. So arcsin(x) answers the question “at what angle does the circle reach height x?” The dashed horizontal line in the diagram is that height, and the vermilion segment is the sine.

§2Domain and range

The domain of arcsin is [−1, 1]. The unit circle has radius 1, so no point on it is higher than 1 or lower than −1, and there is no angle with sine 2. That is why the calculator rejects values outside the interval.

The range is [−90°, 90°], or [−π/2, π/2]. Many angles share the same sine (30° and 150° both have sine 1/2), so to make inverse sine a function, mathematicians agreed to return only the angle on the right half of the circle. That single answer is called the principal value.

§3Exact values of arcsin

Notice that arcsin is odd: arcsin(−x) = −arcsin(x). Negative inputs give negative angles, below the x-axis.

§4All solutions of sin θ = x

Equations usually ask for every angle, not only the principal value. The circle reaches height x at two points in each turn: once at θ₀ = arcsin(x) and once at its mirror image 180° − θ₀. Adding full turns gives all solutions:

In [0°, 360°) that means 30° and 150°. The interactive unit circle shows both points at the same height.

§5sin⁻¹ notation

Calculators label the key sin⁻¹. It means inverse sine, not 1/sin x. The reciprocal 1/sin x is a different function, cosecant (csc x). Writing arcsin avoids the confusion.

§6Finding an angle in a right triangle

In a right triangle, sin θ = opposite / hypotenuse. When you know those two sides, arcsin gives the angle. A ladder 13 feet long that reaches 12 feet up a wall makes an angle of arcsin(12/13) ≈ 67.38° with the ground. Because the angles of a right triangle are always between 0° and 90°, the principal value is exactly the answer you need here.

§7Common mistakes

• Expecting arcsin(sin θ) to give back θ. arcsin(sin 150°) = arcsin(1/2) = 30°, because 150° is outside the range.
• Forgetting the second solution 180° − θ₀ when solving an equation.
• Reading a degree answer in radian mode: arcsin(0.5) is 30 in degree mode and about 0.5236 in radian mode.
• Entering a value like 1.2 that comes from a rounding error; check the earlier step.

§8FAQ