Trig inverse functions

Question

Why doesn't inverse tan of tan (124)=-56 degrees instead of 124 degrees? Doesn't the inverse function cancel the tan function? I am confused.

Answer 1

If you look at a graph of $\tan(x)$, you'll see lots of "up and down" lines, which means the function is not actually invertible - it fails the horizontal line test. To get an inverse, we have to choose one of those lines and flip it across $y=x$. Mathematicians have agreed to choose the part of $\tan(x)$ which goes through the origin (the part for which $-\frac{\pi}{2} < x < \frac{\pi}{2}$, or in degrees, $-90<x<90$). Therefore, the graph of $\tan^{-1}(x)$ only takes values between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ radians, or $-90$ and $90$ degrees. Because of this, $\tan^{-1}(\tan(x))$ is not always equal to $x$.

This means you need to do some thinking when you find an angle by taking $\tan^{-1}$. You have to know whether you are looking for an angle in the first, second, third, or fourth quadrant. The $\tan^{-1}$ function will only give you answers in the first and fourth quadrant, but since $\tan(x+\pi)=\tan(x)$, you can recover answers in the second/third quadrant by adding or subtracting $\pi$ to/from the answer you get. (If you insist on working in degrees, then use $180$ instead of $\pi$.)

Answer 2

The range of inverse tangent is from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. It only returns angles in the fourth and first quadrant.