How to use the Inverse Trig Calculator
- Enter the ratio value you have measured or calculated.
- For arcsin and arccos the value must lie between −1 and 1.
- Arctan accepts any real value, including very large ones.
- Read the angle in degrees from the relevant row.
- Remember each function returns one principal angle; other solutions may also be valid.
How the calculation works
Inverse trigonometric functions run the process backwards: given a ratio, they return the angle that produces it. Because the ordinary functions repeat forever, their inverses must be restricted to a principal range to be single-valued. Arcsin returns −90° to 90°, arccos returns 0° to 180°, and arctan returns −90° to 90° exclusive. Those conventions are universal across mathematics and software.
The restriction has practical consequences. Arcsin(0.5) returns 30°, but 150° also has a sine of 0.5, and in a physical problem the obtuse answer may be the correct one. Whenever you invert a sine, ask whether the supplement is also plausible before proceeding. Arccos has no such ambiguity within a triangle, which is one reason the law of cosines is preferred for angle recovery.
Domain limits are strict for arcsin and arccos: no real angle has a sine or cosine beyond ±1. A value of 1.2 means an error upstream, usually a ratio formed with the wrong side as the denominator. Dividing by a leg instead of the hypotenuse. Arctan, by contrast, is defined everywhere, approaching ±90° asymptotically, which is why slope-to-angle conversions always use it.
θ = arcsin(x) ∈ [−90°, 90°]; arccos(x) ∈ [0°, 180°]; arctan(x) ∈ (−90°, 90°)Source: Principal-value conventions for inverse circular functions; see NIST Digital Library of Mathematical Functions §4.23.
Worked example
A wheelchair ramp rises 0.15 m over a 1.80 m run. Does it meet a 1:12 maximum gradient?
- Slope ratio: 0.15 / 1.80 = 0.0833.
- Angle: arctan(0.0833) = 4.76°.
- The 1:12 limit is also arctan(1/12) = 4.76°.
- The ramp is exactly at the limit, with no tolerance for construction error.
4.76°. Precisely 1:12, so any build inaccuracy would push it out of compliance. Design to 1:14 for margin.
Frequently asked questions
Why do arcsin and arccos reject values above 1?+
Sine and cosine are coordinates on a unit circle and can never exceed its radius. A ratio above 1 means the sides were divided the wrong way round.
What is atan2 and why does software use it?+
It takes the two coordinates separately rather than their ratio, so it can identify the correct quadrant across the full 360°. Plain arctan cannot distinguish a point from its opposite.
Is arcsin the same as 1/sin?+
No. Arcsin is the inverse function; 1/sin is the cosecant. The notation sin⁻¹ means the former despite looking like the latter.
How do I get the other valid angle?+
For sine, subtract the result from 180°. For cosine, negate it. For tangent, add 180°.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.