Question

Why is 0.5[cos(6*PI*x)+cos(8*PI*x)] an alternate form of sin(pi*x)*sin(7*pi*x)?

I know it has something to do with the trigonometric identity
cos(a ± B) = cos(a)cos(B) ∓ sin(a)sin(B)
But I dont understand how it is calculated, please guys!

for example sin(PI*x)sin(7*PI*x) , i can do cos(PI*x+7PI*x) to get cos(8*PI*x), but then how is the cos(a)cos(B) going to become cos(6*PI*x) and where do the 0.5 come from?

Answer 1

I think that you may have to use a multiple-angle formular as well, since you have the coefficients inside on the x. I don't have paper right in front of me, so I can't work it out myself right now, but just try with different identities, sum & difference, and maybe some angle formulas.

Answer 2

I am Norwegian, what is a "multiple-angle formula"?

Answer 3

It is a way to break up the coefficients on the inside next to the x. Here, I'll attatch them. I can't easily explain how it works for angles larger than 2, but for an odd number you can do it like (sine6*pi*x + 1*pi*x) and then use the sum & difference. Sorry I can't be very helpful right now. :/

Answer 4

Ah.... But... So I am supposed to try every possible combination of these until something works?

Answer 5

I would start with sum and difference, then either distribution or multiple angle.

Answer 6

The exercise gives this identity as a hint: cos(a ± B) = cos(a)cos(B) ∓ sin(a)sin(B)

So it is basically just a trick to throw me off?

Answer 7

It would be helpful if someoine calculated it so I could see how it's done properly

Answer 8

Never mind, I found some identities on wikipedia (product-to-sum identities ) that shows it, I dont understand why, but at least some identity shows how it's done... Thanks

Answer 9

The thing to mainly look for here is the fact that 1+7=8 and 7-1=6 since you see both 8 and 6 appear as these kind of oddball things. But once you figure that out, it should sort of slide easily.