Inverse Trigonometric Functions and the Degree-Radian Mode Trap

Sine, cosine, and tangent take an angle and give you a ratio. Inverse sine, inverse cosine, and inverse tangent (written sin⁻¹, cos⁻¹, tan⁻¹, or arcsin, arccos, arctan) run that backward: feed in a ratio, get an angle. The catch is that infinitely many angles share the same sine or tangent value, so the inverse function has to pick one and commit to it. Inverse tangent always returns an angle strictly between -90 and 90 degrees. Inverse sine also sticks to that same narrow band. Inverse cosine returns an angle between 0 and 180 degrees instead. These restricted ranges exist so the inverse functions actually behave like functions, giving one output per input.

Mode matters because the calculator does not know whether the number you typed means degrees or radians. It just applies the trig function to that raw number. In degree mode, sin(30) treats 30 as 30 degrees and returns 0.5. In radian mode, that same 30 is read as 30 radians, a rotation of nearly 5 full circles, which lands the sine value somewhere completely different. Inverse trig functions suffer the same trap in reverse: cos⁻¹(0.5) returns 60 in degree mode but roughly 1.047 in radian mode, since both describe the same angle, just measured on different scales.

This sits right next to the unit circle and the identities around it. Every point on the unit circle is (cos(angle), sin(angle)), so inverse cosine and inverse sine are really asking where on that circle a given coordinate lives. Since both coordinates never leave the range from -1 to 1, any ratio outside that range, like 2, cannot be a sine or cosine of any real angle, so its inverse sine does not exist.