# Thread: sine and cosine rule?

1. ## sine and cosine rule?

Hi I know the formula

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

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

2. Originally Posted by zizou1089
Hi I know the formula

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

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

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

Tonio

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

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

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

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

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

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

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

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

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

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

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

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

If n=1, this is

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

If n=2, it's

$\displaystyle cos(\alpha{\pi})$

so the power of $\displaystyle -1$ is n