Question

How do you solve `\cos 2x = - 0.9`?

Answer 1

Use the inverse cosine. `x=1/2cos^-1(-0.9) +-pin`

Explanation:

To isolate `x`, we can start by taking the inverse cosine of both sides, given by `cos^-1` or `arccos`. You can think of the inverse cosine undoing the cosine function like taking the natural log of `e^x` gets us back to `x` if it helps.

`cos(f(x))=a=>f(x)=arccos(a) color(grey)(+2pin)`

Given `cos(2x)=-0.9`:

`2x=cos^-1(-0.9)+2pin`

Now divide by `2`:

`x=1/2cos^-1(-0.9) +-pin`

We use the period to find the full set of all solutions. The period of `cos(x)` is `2pi`, so these solutions repeat every `2pi` units.