I have two questions:
1) When we assume $n = k$ true, what is the restriction on integer k? I have been told k does not include the first case of n we tested for i.e. k > 1 which makes sense as n = k is an ASSUMPTION and n = 1 is already true.
2) How do I go about the (main) part of the induction proof? After the assumption, we want to show truth to $n = k + 1$. We have LHS: $\sin (x + 180(k+1))$ and I just can't manipulate it to get what I want - it looks very messy using double angles.
Hint: Remember the fundamental period of the cosine and sine functions, and use the identities $$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$ $$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$$
Do you know the interpretation of ($cos\theta , sin\theta$) in terms of coordinates in the unit circle? Basically, $\sin \theta$ is the y-coordinate. Do you see what happens to this coordinate when you rotate by $\pi$ ? This is the antidote, er, I mean antipode; you can obtain $sin(\theta+\pi)$ from $sin\theta$ by drawing a line thru ($cos\theta, sin\theta$) thru the origin.
I got the answer: sin (x + 180(k+1) = sin (x + 180k + 180) From here I used triple angle formula (or double angle formula twice), which was very tedious (as I had to derive it on the spot)
Is there not a cleaner method?