Question

How do you find the general solutions for `2 cos x + 1 = cos x`?

Answer 1

`x= (2k+1)pi`

Explanation:

Subtract `cos(x)` from both sides:

`2cos(x) +1 -cos(x)=cos(x)-cos(x)`

and summing things up, we have

`cos(x)+1=0`

Now subtract one from both sides:

`cos(x)+1-1=0-1`

and so

`cos(x)=-1`

And since the cosine is the projection of the points of the unit circle on the `x`-axis, the only possible point is the "east pole" of the cirle, which means `x=180^@`, or, in radians, `x=\pi`.

Of course this solution is not unique, since you can do as many additional laps as you like, obtaining all the possible solutions

`x=pi+2kpi = (2k+1)pi`