# More Inversion of Trigonometric Equations

Can anyone invert

$$y = a \cos(x) + b \sin(2x)$$

to give $$x = f(y)$$? An exchange on 29 June 2017 said this was possible but I cannot find the solution. Also, is

$$y = a \sin(x) + b \sin(2x)$$

invertible in the same way?

Many thanks.

• Consider using Weierstrass substitution formulas Apr 10 at 21:48
• Where was the "exchange on 29 June 2017"? Can you link to it in the question? Apr 10 at 23:24

Notice that the inverse trigonometric functions are not actually the inverse of trigonometric functions. For example, $$\sin x$$ doesn't actually have an inverse as it fails the horizontal line test. The inverse only exists when we restrict $$\sin x$$'s domain to be $$\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$$.
In your function $$f(x)=a\cos x+b\sin 2x$$, the domain is $$\mathbb{R}$$, and it's a periodic function, so it clearly does not have an inverse function. The same goes with $$g(x)=a\sin x + b\sin 2x$$.