## A sticky trig question

A colleague of mine ran into an issue in the middle of an applied problem. He asked me for help and I wasn’t able to make headway either, so I have obtained permission to open things up to you, my clever readers.

Given x restricted between $\frac{\pi}{2}$ and $\frac{3 \pi}{4}$, find an explicit expression for the inverse of:

$f(x)=\frac{cos(2x)}{cos(x)}$

No identity either of us have come up with makes any headway. Clearly the equation in general doesn’t have an inverse, but it seems like our domain is restricted enough to get something other than an approximation.

### 3 Responses

1. y = cos(2x)/cos(x)
y cos(x) = 2cos^2 (x) – 1
2 cos^2 (x) – y cos(x) – 1 = 0

Use the quadratic formula, except to solve for cos(x), and you get:

cos(x) = (y +/- sqrt(y^2 + 8))/4

So an inverse function could be

f(x) = arccos((x +/- sqrt(x^2 + 8))/4)

I believe if you use the +, you get the piece of the graph from 0 to pi/2, and if you use the -, you get the piece from pi/2 to pi.

2. Thank you!

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