The Arcsin Calculator computes the inverse sine (arcsine) of any value in the domain [−1, 1], returning the principal value angle in degrees and radians. Used in trigonometry, physics, and engineering wherever an angle must be recovered from its sine value — from right triangles to Snell's law.
30
°
0.523599
rad
30
°
0.523599
rad
You know the sine of an angle. You need the angle. That is the arcsin function — the inverse of sine — and the calculator for arcsin computes sin⁻¹(x) for any input in the valid domain [−1, 1], returning the principal value in both degrees and radians. It is one of the most frequently used inverse trigonometric functions in applied mathematics, physics, and engineering.
The arcsine function is the inverse of the sine function, restricted to its principal value range:
arcsin(x) = θ such that sin(θ) = x and −90° ≤ θ ≤ 90°
The domain is −1 ≤ x ≤ 1. Key reference values:
Note that arcsin returns angles in [−90°, 90°] only — it cannot distinguish between an angle and its supplement (since sin(30°) = sin(150°) = 0.5). Context determines which quadrant applies in practical problems. The arccos calculator provides the complementary inverse cosine function.
In a right triangle with opposite side a, adjacent side b, and hypotenuse c, the arcsin directly recovers angle A opposite to side a:
A = arcsin(a/c) — angle from opposite side and hypotenuse
For a right triangle with opposite = 3 m and hypotenuse = 5 m: A = arcsin(3/5) = arcsin(0.6) = 36.87°. Combined with the angle sum A + B = 90° (since the third angle is 90°), all angles are determined from two sides. The arcsin is the appropriate inverse trig function whenever the known ratio involves the opposite side and hypotenuse — the sine ratio.
Arcsin appears prominently in two fundamental physics equations:
The unit circle calculator provides the complete trigonometric values for standard angles, and the trigonometry calculators cover the full inverse trig and angle analysis toolkit.
A beautiful identity connects the two principal inverse trig functions: arcsin(x) + arccos(x) = π/2 = 90° for all x in [−1, 1]. This follows from the complementary angle relationships — since sin(θ) = cos(90° − θ), inverting both sides gives arcsin(x) = 90° − arccos(x). This identity is useful for converting between the two functions and for verifying calculator results: if your arcsin and arccos values for the same input do not sum to exactly 90°, there is a calculation error.
Enter a value between -1 and 1. The calculator applies the inverse sine function to determine the angle whose sine equals your input. Results are displayed in both degrees and radians simultaneously.
The result represents the unique angle in the range $$[-90°, 90°]$$ whose sine equals your input value. A positive result indicates an angle in the first quadrant (above the x-axis), while a negative result indicates an angle in the fourth quadrant (below the x-axis).
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Since sin(30°) = 0.5, the arcsine of 0.5 is exactly 30° or π/6 radians.
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Since sin(−45°) = −√2/2 ≈ −0.707, the result is −45° or −π/4 radians.
They are the same function. The notation $$\sin^{-1}(x)$$ and $$\arcsin(x)$$ both represent the inverse sine function. Note that $$\sin^{-1}(x)$$ does not mean $$\frac{1}{\sin(x)}$$ — that would be the cosecant function $$\csc(x)$$.
The sine function only produces output values in the range $$[-1, 1]$$. Since arcsine is the inverse of sine, its input (domain) must be restricted to this same range. Any value outside $$[-1, 1]$$ has no real angle whose sine equals that value.
$$\arcsin(0) = 0°$$ or $$0$$ radians. This is because $$\sin(0°) = 0$$.
Multiply the radian value by $$\frac{180}{\pi}$$. This calculator displays both units automatically. For example, $$\frac{\pi}{6}$$ radians equals $$30°$$.
No. The principal value of arcsine is restricted to $$[-90°, 90°]$$. If you need the supplementary angle (between 90° and 180°), compute $$180° - \arcsin(x)$$, since $$\sin(\theta) = \sin(180° - \theta)$$.
The derivative is $$\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1 - x^2}}$$, valid for $$|x| < 1$$. The derivative is undefined at $$x = \pm 1$$ because the function's slope becomes vertical at those endpoints.
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