Chapter 1: Trigonometry

1.2 Inverse trig functions

An inverse function can be thought of as a function that “undoes” another function. For example, subtraction undoes addition:

\[ 3 + 5 – 5 = 3 \]

and division undoes multiplication:

\[(3 \cdot 5)\left(\frac{1}{5}\right) = 3 \]

In the same way, there are inverse trig functions.* They are called inverse sine \(\left(\sin^{-1}\right)\), inverse cosine \(\left(\cos^{-1}\right)\), and inverse tangent \(\left(\tan^{-1}\right)\). Each inverse undoes the action of a trig function. For example,

\[\sin^{-1}\left(\sin\left(42^\circ\right)\right) = 42^\circ\]

Here, the sine function does something to the angle 42° (specifically, the sine of 42° is 0.669). Then the inverse sine function undoes whatever sine did, resulting in 42°. Try it yourself: use a calculator to first take the sine of some angle, then take the inverse sine of that result; you’ll see that you end up back where you started.

OK, so this is neat…but why is it useful?

If you know two sides and need to find out the angle, you can use an inverse trig function. For example, say you know sides \(a\) and \(c\) of the triangle in the figure above. You could find the angle \(\theta\) by taking the inverse sine of both sides of the equation:

\sin(\theta) &= \frac{a}{c} \\
\sin^{-1}\left(\sin(\theta)\right) &= \sin^{-1}\left(\frac{a}{c}\right)

On the left hand side, we take the inverse sine of the sine of the angle; this “undoes” the operation of sine:

\[ \theta = \sin^{-1}\left(\frac{a}{c}\right) \]

Given values for \(a\) and \(c\), we could enter the inverse sine of \(a\) divided by \(c\) into the calculator and find a numerical result for the angle. Note that the inverse trig function is itself a function: it has some input and it gives an output. On most scientific calculators, you access the inverse trig functions by first hitting the 2nd or “shift” or ↑ button, then the button for the trig function you want to take the inverse of.

Practice 1.5

*As with much of the mathematics that we use in physics, there’s a bit more nuance to which functions have an inverse, and how that inverse is defined. You don’t really need those details to simply use the inverse trig functions like we’ll do here. You’ll learn all the fun details and nuances as you take more math classes.