Arcsine Calculator
What Inverting the Sine Means
An arcsine calculator answers a reverse question: which angle has this sine value? You provide a ratio between -1 and 1, and the tool returns the angle whose sine equals your input. Students untangle trigonometry homework with it, and engineers use it to recover an angle of incline from a measured rise and slope length.
It is the exact inverse of the sine function on its principal branch. That branch restriction keeps every returned answer unique.
The Formula and a Worked Example
The function is written as arcsin(y), where y = opposite / hypotenuse. Outputs are principal values restricted to the range from -90 to 90 degrees. Anchors worth memorizing: arcsin 0.5 = 30 degrees, arcsin 1 = 90 degrees, and arcsin 0 = 0 degrees.
Picture a ramp that rises 1 meter over a 2 meter slope length. The ratio is 1 / 2 = 0.5, so the calculator returns arcsin 0.5 = 30 degrees. The ramp climbs at thirty degrees, and doubling the ratio to 1 would push the answer to the maximum, 90 degrees.
How to Use It
Enter your ratio into the Sine box and press Calculate. A ramp rising 1 meter over 2 meters of slope corresponds to an input of 0.5, and the answer comes back as 30 degrees. Any similar rise over hypotenuse measurement follows the same pattern.
Your input must lie between -1 and 1; a value such as 2 has no answer because no angle has a sine that large. Remember the output is the principal angle: sin 150 degrees is also 0.5, yet the calculator reports 30 degrees, the one angle inside the -90 to 90 degree window.
Frequently Asked Questions
What is arcsin 0.5 in degrees?
Arcsin 0.5 is exactly 30 degrees, since sin 30 = 0.5. The alternative angle 150 degrees also has sine 0.5, but it sits outside the principal range from -90 to 90 degrees.
Why is the range -90 to 90 degrees?
Sine repeats its values across many angles, so restricting arcsin to principal values from -90 to 90 degrees guarantees one unique answer. Without that limit, arcsin 0.5 could return 30 or 150 or 390 degrees.
What happens if I enter 1.5?
The calculator cannot return an angle, because every sine value is trapped between -1 and 1. Inputs from -1 to 1 are valid, and arcsin 1 returns exactly 90 degrees.
What is arcsin of -1?
Arcsin(-1) equals -90 degrees, the lowest angle the principal range allows. That angle points straight downward, since a line descending 1 unit over a hypotenuse of 1 unit makes a right angle with the horizontal.
Is arcsin the same as cosecant?
They are different functions: arcsin inverts sine and returns an angle, while cosecant is 1 / sin and returns a ratio. For example, arcsin 0.5 = 30 degrees, but csc 30 = 2.