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Arcsine Principal Angle & Sine Ratio

Find the principal inverse sine angle from a ratio, with negative angles, reverse entry and degrees, radians, turns or arc units.

Calculator inputs

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Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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The principal inverse sine of −0.5 is −30°, or −π/6 radians. The principal inverse sine of 0.5 is 30°.

Angle = arcsin(ratio); ratio = sin(angle), with the principal angle from −π/2 to π/2 radians

Enter these known values and leave the other values blank.

The real-valued sine ratio must be between −1 and 1. The principal result is between −90° and 90°, inclusive.
Reverse angle entry uses this same principal branch, with numerical conversion tolerance at the boundaries.
A negative sine ratio produces a negative principal angle. A coterminal angle such as 330° is not the principal arcsine result.
The inverse function notation sin⁻¹ means arcsine here, not the reciprocal sine function.

Both angles have the same sine, but arcsine returns the angle within its defined principal range of −90° to 90°.

Bookify restricts reverse angle entry to the principal inverse-trigonometric branch, with a 1e-50-radian tolerance for unit-conversion rounding.