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If you build a wheelchair ramp with a 1-foot rise and a 12-foot run, how steep is the climb?
Standard trigonometry uses an angle to calculate side lengths, but when you already have the sides, you need a way to run the process backward.

Running Trig in Reverse

An inverse function is an operation that undoes another operation, just like taking a square root undoes squaring.
We write these inverse operations as sinβ‘βˆ’1\sin^{-1}, cosβ‘βˆ’1\cos^{-1}, and tanβ‘βˆ’1\tan^{-1}, also known as arcsine, arccosine, and arctangent.
In 1772, mathematician Joseph-Louis Lagrange popularized the 'arc-' prefix because finding the inverse trig value meant finding the arc length on a unit circle corresponding to a given ratio.
Be careful with the βˆ’1-1 symbol: here it denotes an inverse function, not an algebraic exponent, meaning sinβ‘βˆ’1(x)β‰ 1sin⁑(x)\sin^{-1}(x) \neq \frac{1}{\sin(x)}.
How do you know which of the three inverse buttons to press? It all depends on which two sides you know.

Choosing the Right Ratio

Every right triangle names its sides relative to an acute angle θ\theta: the hypotenuse is opposite the 90∘90^\circ angle, the opposite side faces θ\theta, and the adjacent side touches θ\theta.
What does this look like in a real calculation? Let's return to our ramp problem.