Solve $\cos 2x = \frac{2x}{3}$

Question

I'm trying to solve for $x$ in the following equation: $$\cos 2x = \frac{2x}{3}$$ Any help would be much appreciated!

Answer 1

This is a transcendental equation, so it doesn't have any "nice" solutions - typically, one investigates solutions by drawing the graphs of $f(x)=\cos(2x)$ and $g(x)=\frac{2}{3}x$ and noting where they intersect, or use numerical methods.

You could write $\cos(2x)=\cos^2(x)-\sin^2(x)$ but this doesn't get you any closer to a solution.

See the Wolfram Alpha page for the solutions and the graph of $f$ and $g$.