Inverse Sine Calculator
Find the angle whose sine equals a given value using the arcsine (sin⁻¹) function. Use this when solving triangles, physics problems, or any scenario where you know a sine ratio and need the corresponding angle.
About this calculator
The inverse sine function, written as arcsin or sin⁻¹, answers the question: 'What angle has this sine value?' If sin(θ) = x, then θ = arcsin(x). The input value must lie between −1 and 1 (inclusive), because the sine function never produces values outside this range. The output angle falls in the range −90° to 90°. The formula applied here is: θ (degrees) = arcsin(value / 100) × (180 / π), where the value is entered as a percentage of 1 (i.e., the raw sine ratio multiplied by 100). This is the principal value returned by most calculators and programming languages. Arcsine is essential in trigonometry, physics (projectile motion, wave analysis), and engineering whenever an angle must be recovered from a known ratio of opposite to hypotenuse.
How to use
Suppose you know that the sine of an unknown angle is 0.5 (enter value = 50, since the field accepts the ratio × 100). Step 1: The calculator computes arcsin(50 / 100) = arcsin(0.5). Step 2: arcsin(0.5) = 30° (since sin(30°) = 0.5). So the result is 30°. For another example, enter value = 70.71 to get arcsin(0.7071) ≈ 45°, confirming the 45° triangle relationship. Values outside −100 to 100 will return an error.
Frequently asked questions
What values can I enter into the inverse sine calculator?
You may enter any number between −100 and 100 (representing sine ratios from −1 to 1, scaled by 100). Values outside this range are mathematically invalid because the sine function is bounded between −1 and 1, and no real angle produces a sine value beyond these limits. Entering a value like 150 will return an error. If your raw sine ratio is 0.866, enter 86.6 into the calculator to get the corresponding angle of approximately 60°.
How is arcsine different from sine, and when should I use each?
Sine takes an angle as input and returns a ratio (opposite / hypotenuse), while arcsine takes a ratio and returns the corresponding angle. Use sine when you know the angle and need the ratio; use arcsine when you know the ratio and need the angle. For example, in a right triangle if you know two sides but not the angle, you would compute the ratio and then apply arcsine to find the angle. Both functions are inverses of each other within their defined domains.
Why does arcsine only return angles between −90° and 90°?
The sine function is not one-to-one over all angles — many different angles share the same sine value. For example, both 30° and 150° have sin = 0.5. To define a proper mathematical inverse, mathematicians restrict the output to the principal value range of −90° to 90°, where sine is strictly increasing and one-to-one. This is the value your calculator returns. If you need the other possible angle (e.g., 150° instead of 30°), you can compute it as 180° minus the principal arcsine result.