Inverse Sine Calculator

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Enter a value between -1 and 1 (the sine of the angle)

Calculate the inverse sine (arcsin) of any value between -1 and 1. Get the angle in degrees, radians, and gradians with step-by-step verification. Perfect for trigonometry problems.

Arcsin Result

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Sources & Methodology

✓ Formulas verified against authoritative sources listed below.

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NIST Digital Library of Mathematical Functions — Chapter 4: Trigonometric Functions

NIST authoritative reference for inverse trigonometric functions including arcsin definition, range, and properties

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Wolfram MathWorld — Inverse Sine

Comprehensive mathematical reference for arcsine properties, series expansions, and applications

Methodology: arcsin(x) = inverse of sine function. Domain: -1 to 1. Range: -90 to +90 degrees (-pi/2 to pi/2 radians). Computed using JavaScript Math.asin(x). Conversion: degrees = radians x (180/pi). Gradians = degrees x (10/9). Verification: sin(arcsin(x)) = x.

⏱ Last reviewed: April 2026

How to Calculate the Inverse Sine

The inverse sine function (arcsin, also written sin⁻¹ or asin) returns the angle whose sine equals a given value. If sin(30°) = 0.5, then arcsin(0.5) = 30°. Unlike arccos, the arcsin principal value range is −90° to +90° (−π/2 to π/2 radians), meaning it returns negative angles for negative inputs.

The Arcsin Function Defined

arcsin(x) returns the angle θ in the range −90° to +90° such that sin(θ) = x. Key values: arcsin(0) = 0°, arcsin(0.5) = 30°, arcsin(1) = 90°, arcsin(−0.5) = −30°, arcsin(−1) = −90°. Values outside −1 to 1 are undefined because sine never exceeds those bounds.

Arcsin Range vs. Arccos Range

The principal range of arcsin is −90° to +90°. The principal range of arccos is 0° to 180°. This is why: sine is symmetric around 90° (sin(30°) = sin(150°) = 0.5), so arcsin picks the value in the first and fourth quadrants. Cosine picks the first and second quadrant range (0° to 180°). Together: arcsin(x) + arccos(x) = 90° for all x.

Applications of Inverse Sine

Arcsin appears in: 1) Right triangle problems — finding an angle from the opposite side and hypotenuse: A = arcsin(opposite/hypotenuse). 2) Physics — projectile motion angle calculations. 3) Navigation — latitude calculations, bearing problems. 4) Signal processing — phase angle recovery. 5) Computer graphics — inverse kinematics joint angle calculation.

Arcsin Identity: arcsin(x) + arccos(x) = 90 Degrees

This identity holds for all x in −1 to 1: arcsin(x) + arccos(x) = 90° (π/2 radians). This means knowing either arcsin or arccos gives the other by subtraction. For example: arcsin(0.5) = 30°, so arccos(0.5) = 90° − 30° = 60°. This complementary relationship reflects the co-function identity sin(θ) = cos(90° − θ).

arcsin(x) = angle where sin(angle) = x | Domain: -1 to 1 | Range: -90 to +90 degrees (-pi/2 to pi/2 rad)

Conversion: degrees = radians x (180/pi). Radians = degrees x (pi/180). Identity: arcsin(x) + arccos(x) = 90 degrees for all x in [-1,1]. Verification: sin(arcsin(x)) = x always. Common: arcsin(0.5) = 30 deg, arcsin(sqrt(2)/2) = 45 deg, arcsin(sqrt(3)/2) = 60 deg.

Inverse Sine Common Values

x value	arcsin(x) degrees	arcsin(x) radians	Notes
-1	-90°	-π/2 (-1.5708)	sin(-90°) = -1
-0.866	-60°	-π/3 (-1.0472)	sin(-60°) = -sqrt(3)/2
-0.707	-45°	-π/4 (-0.7854)	sin(-45°) = -sqrt(2)/2
-0.500	-30°	-π/6 (-0.5236)	sin(-30°) = -1/2
0	0°	0	sin(0°) = 0
0.500	30°	π/6 (0.5236)	sin(30°) = 1/2
0.707	45°	π/4 (0.7854)	sin(45°) = sqrt(2)/2
0.866	60°	π/3 (1.0472)	sin(60°) = sqrt(3)/2
1	90°	π/2 (1.5708)	sin(90°) = 1

💡 Key Distinction: arcsin returns angles from -90° to +90°. If you know the angle is in the second quadrant (90° to 180°), use 180° minus the arcsin result: if arcsin(x) = 30°, the second quadrant angle with the same sine is 180° - 30° = 150°. This is because sin(30°) = sin(150°) = 0.5. Always consider which quadrant your angle is in when interpreting arcsin results.

Frequently Asked Questions

Inverse sine (arcsin) is the function that returns the angle whose sine equals a given value. If sin(30 degrees) = 0.5, then arcsin(0.5) = 30 degrees. Domain: -1 to 1. Range (principal values): -90 to +90 degrees.

Use the arcsin or sin^-1 button on a scientific calculator. In programming: Math.asin(x) returns radians; multiply by 180/pi for degrees. For arcsin(0.5): Math.asin(0.5) = 0.5236 radians x (180/pi) = 30 degrees.

arcsin(0) = 0 degrees = 0 radians. sin(0 degrees) = 0.

arcsin(1) = 90 degrees = pi/2 radians. sin(90 degrees) = 1.

arcsin(0.5) = 30 degrees = pi/6 radians. sin(30 degrees) = 0.5. This is one of the most common arcsin values to memorize.

arcsin(-1) = -90 degrees = -pi/2 radians. sin(-90 degrees) = -1.

arcsin(x) + arccos(x) = 90 degrees (pi/2 radians) for all x in -1 to 1. So arccos(x) = 90 - arcsin(x). For example: arcsin(0.5) = 30 degrees, arccos(0.5) = 60 degrees. 30 + 60 = 90.

Domain of arcsin: -1 to 1 (sine never exceeds this range). Range (principal values): -90 degrees to +90 degrees (-pi/2 to pi/2 radians). The function is monotonically increasing on this range.

In a right triangle: sin(angle) = opposite / hypotenuse. So angle = arcsin(opposite / hypotenuse). Example: opposite = 5, hypotenuse = 10. angle = arcsin(5/10) = arcsin(0.5) = 30 degrees.

Because sine is not one-to-one over its full domain (sin(30) = sin(150) = 0.5). To define an inverse function, we restrict to the range where sine is one-to-one: -90 to +90 degrees. This is called the principal value range. For other quadrant solutions, use the supplementary angle identity.

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