Trigonometric functions and radians

[Mattara]

Solve the following equation giving values from $$-\pi$$ to $$\pi$$:

$$cos (2v - \frac{\pi}{3}) = \cos v$$

Here is my attempt to solve it.

As the cosine of the two is the same, the angles should also be the same leaving

$$2v - \frac{\pi}{3} = v + 2 \pi n$$

Then if I move the right over to the left, I get one of the correct solutions ($$\frac{\pi}{3}$$) and I know that the others is gotten by using the periodicity and the nature of cosine (being the same for both negative and positive values) although I'm not entirely sure on how to impletemt it.

I think that it the periodicity should be added to the right hand side as above.

Any hints or guidance is highly appreciated. Thank you for your time and have a nice day

 

[StatusX]

cos(x)=cos(y) is true if x=y+2$\pi$n. But, as you point out, it's also true if x=-y, and so also if x=-y+2$\pi$n. This exhausts the solutions.

 

[Mattara]

I managed to figure it out. Thanks :)

Note to self: n can be -1