# Thread: sine and cosine rule?

1. ## sine and cosine rule?

Hi I know the formula

$sin(\alpha \pm n)\pi=(-1)^nsin\alpha \pi$

but what about

$cos(\alpha \pm n)\pi=?$
$cos(\alpha \mp n)\pi=?$
$sin(\alpha \mp n)\pi=?$

2. Originally Posted by zizou1089
Hi I know the formula

$sin(\alpha \pm n)\pi=(-1)^nsin\alpha \pi$ , but what about

$cos(\alpha \pm n)\pi=?$

$cos(\alpha \pm n)\pi=(-1)^n\cos\alpha\pi$ , assuming $n\in\mathbb{Z}$ , and the other two below are, of course, exactly the same as the first two (why do you think they're different??)

Tonio

$cos(\alpha \mp n)\pi=?$

$sin(\alpha \mp n)\pi=?$
.

3. Just making sure about the other 2! Thanks!

4. Originally Posted by tonio
.
would it not be.. $cos(\alpha \pm n)\pi=(-1)^{n+1}\cos\alpha\pi$??

5. Originally Posted by zizou1089
would it not be.. $cos(\alpha \pm n)\pi=(-1)^{n+1}\cos\alpha\pi$??
cos(a+ b)= cos(a)cos(b)- sin(a)sin(b).

In particular, if $b= n\pi$ and $a= \alpha$,
$cos(\alpha+ n\pi)= cos(\alpha)cos(n\pi)- sin(\alpha)sin(n\pi)$.

Of course, $sin(n\pi)= 0$, for all n, and $cos(n\pi)= (-1)^n$.

Therefore, $cos(\alpha+ n\pi)= (-1)^ncos(\alpha)$.

For $cos(\alpha- n\pi)$, use the fact that cosine is an even function: $cos(-n\pi)= cos(n\pi)= (-1)^n$

there is no difference at all between $\alpha\pm n\pi$ and $\alpha\mp n\pi$. They both mean exactly the same thing.

6. Originally Posted by zizou1089
would it not be.. $cos(\alpha \pm n)\pi=(-1)^{n+1}\cos\alpha\pi$??
$cos(\alpha\pm{n}){\pi}=cos(\alpha{\pi})cos(\pm{n}{ \pi})-sin(\alpha{\pi})sin(\pm{n}{\pi})
$

$=cos(\alpha{\pi})cos(\pm{n}{\pi})$

If n=1, this is

$cos(\alpha{\pi})cos(\pm{\pi})=-cos(\alpha{\pi})$

If n=2, it's

$cos(\alpha{\pi})$

so the power of $-1$ is n