I can not find a good way to solve this rather simple-looking equation. cos⁡x+cos⁡2x=2 I...

lugreget9

lugreget9

Answered

2021-12-31

I can not find a good way to solve this rather simple-looking equation. cosx+cos2x=2
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

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 2x are a multiple of 2π. Since 2 is not rational, there is no such multiple. In other words, the only solution is when:
x=2x=0x=0
Cleveland Walters

Cleveland Walters

Expert

2022-01-02Added 40 answers

Use that
cos(x)+cos(y)=2cos(x2y2)cos(x2+y2)
nick1337

nick1337

Expert

2022-01-08Added 573 answers

From 1cosx1,  xR, we get cosx+cos(2x)2. The equality when x=2kπ : (1) and 2x=2λπ, where k,λZ. From (1) we get 2(2kπ)=2λπ222k=2λ, which is imposible when k,λ integers and k0(2 is irrational). Hence cosx+cos(2x)<2,  xR{0}. Hence the given equation has no real roots ecxept for the case x=0.

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