How to write a complete solution set for $x$ in trigonometry questions?

Question

Should I be able to perform this operation? $$\cos 3x=1\Rightarrow3x=\cos^{-1}1\Rightarrow \bbox[yellow,5px]{x=\frac{\cos^{-1}1}3}$$

Well it gives a right answer because $\cos (3\times0)=1$, but I do understand that it does not give all the answers.

Using mathematica, it tells me that the complete set of solutions is: $$x\rightarrow\frac{2\pi}3\cdot C, \ \ \ \mathbb{for \ }C\in\mathbb{Z}$$

Additionally

How do I write the conditional expression that defines all the answer for any trigonometry question? Does it help that I should be working in radians? Or is it solvable when working in degrees to?

Answer 1

General solution of trigonomertic equations:

$\cos x=\cos \theta$ $\implies$ $x=2n\pi\pm\theta$, where $n\in\mathbb{Z}$

$\tan x=\tan\theta$ $\implies$ $x=n\pi+\theta$, where $n\in\mathbb{Z}$

$\sin x=\sin\theta$ $\implies$ $x=n\pi+(-1)^n\theta$, where $n\in\mathbb{Z}$

For this question,

\begin{align*} &\;\cos 3x=1\\ \implies &\;\cos 3x=\cos0\\ \implies &\; 3x=2n\pi\pm 0=2n\pi \quad(n\in\mathbb{Z})\\ \implies &\; x=\frac{2n\pi}{3} \quad(n\in\mathbb{Z})\\ \end{align*}

Answer 2

It is much simpler to use the language of congruences: $$\cos 3x=1\iff3x\equiv 0\mod 2\pi\iff x\equiv 0\mod\frac{2\pi}3.$$

The unit is unrelated to solving or not. This being said, the natural unit, mathematically speaking, is the radian.