lugreget9

Answered

2021-12-31

I can not find a good way to solve this rather simple-looking equation. $\mathrm{cos}x+\mathrm{cos}\sqrt{2x}=2$

I can see that 0 is a solution, but is there a good way of solving it for all the potential solutions.

I can see that 0 is a solution, but is there a good way of solving it for all the potential solutions.

Answer & Explanation

peterpan7117i

Expert

2022-01-01Added 39 answers

You have already found all solutions.

The sum of those cosines can only be 2 if both x and$\sqrt{2}x$ are a multiple of $2\pi$ . Since $\sqrt{2}$ is not rational, there is no such multiple. In other words, the only solution is when:

$x=\sqrt{2}x=0\Rightarrow x=0$

The sum of those cosines can only be 2 if both x and

Cleveland Walters

Expert

2022-01-02Added 40 answers

Use that

$\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)=2\mathrm{cos}(\frac{x}{2}-\frac{y}{2})\mathrm{cos}(\frac{x}{2}+\frac{y}{2})$

nick1337

Expert

2022-01-08Added 573 answers

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