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Math Help - sine and cosine rule?

  1. #1
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    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=?
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  2. #2
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    Quote Originally Posted by zizou1089 View Post
    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=?
    .
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  3. #3
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    Just making sure about the other 2! Thanks!
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  4. #4
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    Quote Originally Posted by tonio View Post
    .
    would it not be.. cos(\alpha \pm n)\pi=(-1)^{n+1}\cos\alpha\pi??
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  5. #5
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    Quote Originally Posted by zizou1089 View Post
    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.
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  6. #6
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    Quote Originally Posted by zizou1089 View Post
    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})<br />

    =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
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