Question

Find the general solution to: cos 5x - cos 2x = 0

Answer 1

start with the identity\[\cos p -\cos q=-2 \sin(\frac{p+q}{2}) \sin(\frac{p-q}{2})\]

Answer 2

Or: write as cos 5x = cos 2x

The general solution of cos a = cos b is: \[a= \pm b +2k \pi\]where k is an integer.

Answer 3

Ok, this might be above my level.

Answer 4

Might I ask where you would find the integer k if you needed too?

Answer 5

I think that doesn't have to be the case.

If you use the general solution to cos a = cos b, you get:\[5x=\pm 2x + 2k \pi\]If this looks difficult, write as two separate equations:\[5x=2x + 2k \pi \vee5x=-2x + 2k \pi\]This means:\[3x=0+2k \pi \vee 7x=0+2k \pi\]Then, it can be written simpler by leaving out the 0:\[3x=2k \pi \vee 7x = 2 k \pi\]Now divide by 3 (first eqn) and 7 (second eqn) to see your solutions!

Answer 6

@Henry.Lister: you don't have to find k. It can be ANY integer! That is why there is an infinity of solutions.

In the image you can see why! (red graph: cos 5x. Blue graph: cos 2x)

Answer 7

Thank you very much.

Answer 8

You're welcome!

Answer 9

BTW, I got x = 2k/3 pi and x = 2k/7 pi as solutions.
You can see these as the intersection points of the two graphs.