Calculate inverse sine (arcsin) values in degrees, radians, and gradians. Find the angle θ from any sine ratio with step-by-step results.
A wheelchair ramp rises 1 meter over a length (hypotenuse) of 12 meters.
Formula: θ = arcsin(opposite / hypotenuse) = arcsin(1/12)
arcsin(0.0833) ≈ 4.78°
Arcsin gives the angle from the opposite side and hypotenuse, useful when only the ramp length is known.
A roof has a rise of 4 meters and a rafter length (hypotenuse) of 7.21 meters.
Formula: θ = arcsin(rise / rafter) = arcsin(4/7.21)
arcsin(0.5548) ≈ 33.69°
Roof pitch angle can be found using the rise and rafter length with the arcsine function.
A projectile is launched at a speed of 50 m/s and reaches a maximum height of 31.9 meters.
Formula: θ = arcsin(√(2gh)/v) = arcsin(√(2 × 9.81 × 31.9)/50)
arcsin(0.5) ≈ 30.00°
Arcsin is used in physics to determine launch angles from height and initial velocity.
A satellite dish is pointed at a satellite. The dish height is 2 meters above ground and the line-of-sight distance to the satellite is 100 meters.
Formula: θ = arcsin(height / distance) = arcsin(2/100)
arcsin(0.02) ≈ 1.146°
Arcsin calculates the elevation angle when the vertical height and slant distance are known.
The arcsine function (arcsin or sin⁻¹) is the inverse of the sine function. For a given value x, arcsin(x) returns the angle θ whose sine is x. In a right triangle, this is the angle whose opposite/hypotenuse ratio equals the input value.
arcsin(x) returns an angle between -90° and 90° (inclusive). For x > 0, the angle is in Quadrant I. For x < 0, it is in Quadrant IV.
arcsin(-x) = -arcsin(x). The arcsine is an odd function, symmetric about the origin.
arcsin(x) is only defined for x in [-1, 1]. Values outside this range are not valid for real-number arcsine, since sine values always lie within [-1, 1].
90° = 100 gradians. arcsin(0.5) = 30° = 33.33 gon. arcsin(1) = 90° = 100 gon. Use gradians for metric-based surveying.
The arcsine function (abbreviated arcsin, sin⁻¹, or asin) is the inverse of the sine function. While the sine function takes an angle and returns a ratio, the arcsine function takes a ratio and returns the corresponding angle. It answers the question: "What angle θ has a sine equal to x?"
In a right triangle, the sine of an angle θ is the ratio of the opposite side to the hypotenuse: sin(θ) = opposite/hypotenuse. Therefore, arcsin(opposite/hypotenuse) = θ. This makes arcsine essential for determining angles when you know the side lengths of a triangle, particularly when you have the opposite side and the hypotenuse.
The arcsine function has a domain of [-1, 1] and a range of [-π/2, π/2] radians (or -90° to 90°). Any sine value must fall within [-1, 1], and the arcsine will always return a principal value within this range. Arcsine is an odd function, satisfying arcsin(-x) = -arcsin(x).
Calculating arcsine values can be done in several ways. The most direct method is using the inverse trigonometric functions available on scientific calculators or in programming languages (asin, Math.asin, or arcsin). For common values, the arcsine can be derived from known sine ratios.
Method 1 — Direct Calculation: Enter the sine value x (must be between -1 and 1) into a calculator and press the sin⁻¹ or arcsin function. For example, arcsin(0.5) = 30° because sin(30°) = 0.5.
Method 2 — Ratio Method: If you know the opposite side and hypotenuse of a right triangle, divide the opposite by the hypotenuse to get x, then apply the arcsin function. For example, a ramp that rises 1 meter over a 2-meter hypotenuse has an angle of arcsin(1/2) = arcsin(0.5) = 30°.
Method 3 — Special Values: For the common angles (0°, 30°, 45°, 60°, 90°), the sine values are exact: sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2 ≈ 0.7071, sin(60°) = √3/2 ≈ 0.8660, sin(90°) = 1. The arcsine of these values gives the corresponding angle.
These exact values are the building blocks of inverse trigonometry:
The arcsine function may seem like a purely mathematical concept, but it has numerous practical applications that affect our everyday lives. Understanding arcsine helps in fields ranging from construction to robotics, from navigation to game development.
Building codes specify maximum slopes for ramps. When you measure the rise and the actual ramp length (hypotenuse), arcsine converts these measurements into the angle of incline for compliance checking.
In sports analytics, arcsine helps calculate launch angles for projectiles. When you know the vertical height achieved and the launch speed, arcsine determines the optimum angle for maximum distance.
In game development, arcsine is used in 3D environments for calculating vertical camera angles, field-of-view adjustments, and character aiming mechanics involving elevation.
Pilots use arcsine to calculate glide slopes and descent angles. Given the altitude and the slant distance to the runway, arcsine provides the descent angle for a safe approach.
⚠️ Important Note: This Arcsine Calculator is for educational and informational purposes only. While every effort has been made to ensure accuracy, results should be verified independently for critical applications such as engineering, construction, navigation, or medical devices. Always consult a qualified professional for decisions involving trigonometric calculations in high-stakes contexts.