Trigonometry

This going to sound dumb, but say if I'm given something like cos(nθ, is there any way at all to get the n out?

You see, I'm stuck on my math assignment. It looks like I've almost got it, but I think I may have driven myself into a corner because I have to do the above after reducing powers.

I've never been good at proving trigonometric identities and stuff.


EDIT: Well, at least I'll get half-credit for this assignment. I can do the other half of it, which is to prove the exact same identity with a different method.

:>

The rest is given as an exercise for the reader.

There's also:

 Spoiler

This is basically my way of saying that what you're trying to achieve is best done through algebra or rethinking the problem. There's no sane way of removing the n out of cos(n*theta) without it resulting in annoyance.

Okay, I give up.



I have to prove it via mathematical induction.

Which means I have to make this:



equal to this:



right?

Or is it easier to start from the left instead?

p.s. Don't work it out for me; I'll try again when I'm less stressed out.

EDIT: Wait, wait. Hold that thought. WHAT WAS I EVEN DOING ANYWAY. I don't need to equate those two. I need to equate the sum from 1 to k plus the (k+1)th term with the sum to (k+1) using the equation on the other side.

Great. Now I look like an idiot.

EDIT2: Nope, still having problems. I've proven the base case for the conjecture, but the inductive step is proving to be as much of a pain as what I was initially doing.

To prove by induction:

Prove base case for n=1

Assume n=k case is true

Start with the n=k+1 case's left side (or right side)... and somehow use the n=k case to get the right side (or left)

edit: Since you asked for it not to be solved, let us know how you do and if you need any other hints.

I think I wrote myself into a corner. Again.

I have no idea how I'm gonna equate

k(cos(kθ + cosθ + cos(kθcosθ - sin(kθsinθ

to

(sin(k + 3/2)θ - sin(θ/2))/2sin(θ/2)

Either I screwed up somewhere or I did something unnecessary and screwed myself over.

Start again, right from the start. I suspect you made serious careless mistakes too early on.

And don't hurry to combine numbers. For math problems, those tend to make you unable to see critical terms.

Hadriel

Okay. Freaking biggest mistake of my life.

I was too stuck on what was taught in class and just assumed it was an arithmetic progression.

It's not.

It's not a geometric progression either.

Back to the drawing board. I think I should be able to do it now.

tl;dr I'm a fucking idiot.

Always ready to help. You can tell us the general idea to see if you're on the right track. We won't necessarily tell you the right answer, but we can tell you where is wrong, or what is flawed thinking. [That's what educators are for.]

Hadriel

Always ready to help. You can tell us the general idea to see if you're on the right track. We won't necessarily tell you the right answer, but we can tell you where is wrong, or what is flawed thinking. [That's what educators are for.]

Hadriel

Thanks, but I finally managed to solve it.

I went with P(k-1) and P(k) instead of P(k) and P(k+1) so I wouldn't have to deal with cos(k+1)θ.

Thanks, but I finally managed to solve it.

I went with P(k-1) and P(k) instead of P(k) and P(k+1) so I wouldn't have to deal with cos(k+1)θ.

Definitely an excellent variation on proofs by induction. You'll just have to be careful with naming.... because the k=1 case becomes P(0) and P(1) instead of P(1) and P(2) like before.

Definitely an excellent variation on proofs by induction. You'll just have to be careful with naming.... because the k=1 case becomes P(0) and P(1) instead of P(1) and P(2) like before.

Well, r>0 so I can't use P(0).