Finding an Angle With Inverse Trig

Regular trig functions take an angle and give you a ratio of sides. Sine of 30 degrees gives you 0.5. But often on a right triangle problem you know two sides and need the angle itself, and there's no angle to plug in yet. That's a different direction of travel, so it needs a different button: sin⁻¹, cos⁻¹, or tan^−1. These are the inverse functions, and each one undoes its regular counterpart. If tan(theta) = opposite/adjacent, then theta = tan⁻¹(opposite/adjacent). You're asking the calculator: what angle has this ratio?

Picking the right inverse function works exactly like picking the right ratio normally: look at which two sides you know relative to the angle you want. Know opposite and adjacent? Use tan^−1. Know opposite and hypotenuse? Use sin^−1. Know adjacent and hypotenuse? Use cos^−1. The calculator then hands back a degree measure directly, because that's what the inverse function is built to output: an angle, not a ratio.

This sits right next to the regular trig ratio skill and the two special triangles — once you can set up opposite, adjacent, and hypotenuse correctly for sine, cosine, and tangent, inverse trig is the same setup run in reverse. It also connects forward to the Law of Sines and Law of Cosines, which use these same ratios and their inverses on triangles that don't have a right angle at all.