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Thread: Trigonometric equations

  1. #1
    Member roshanhero's Avatar
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    Trigonometric equations

    In my book following result is shown but don't know where it come from.
    1.2cos^2(2kpie)=2 (where k is an integer)
    2.cos(kpie)=(-1)^k
    3.[sin(kpie)cos(theta)-cos(kpie)sin(theta)]=[(-1)^k-1 sin(theta)]
    In all these cases I just don't know how the results came.Please explain it to me.
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  2. #2
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    Hi roshanhero,

    1) The cosine function is periodic with period $\displaystyle 2\pi $ and thus $\displaystyle \cos (2k\pi)=\cos (2\pi)~\forall k \in \mathbb{Z}$ , hence $\displaystyle 2\cos^2 (2k\pi)=2\cos^2 (2\pi)~\forall k \in \mathbb{Z}$ The conclusion follows.

    2) If $\displaystyle k$ is even then $\displaystyle k=2n$ for some $\displaystyle n$ and hence $\displaystyle \cos (k\pi)=\cos (2n\pi)=\cos (2\pi)~\forall n \in \mathbb{Z}$ (which follows from above). Notice that $\displaystyle (-1)^k=[(-1)^2]^n=1^n=1~\forall n \in \mathbb{Z}$ Thus if $\displaystyle k$ is even then the statement is true since $\displaystyle \cos (2\pi)=1$ Note that due to the cosine function's periodic attributes $\displaystyle \cos (\psi+\pi)\equiv-\cos (\psi)$ . If $\displaystyle k$ is odd then $\displaystyle k=2r+1$ for some $\displaystyle r$ and hence $\displaystyle \cos (k\pi)=\cos [(2r+1)\pi]=\cos (2r\pi+\pi)=-\cos (2r\pi) = -\cos (2\pi)~~\forall r \in \mathbb{Z}$ Notice that $\displaystyle (-1)^k=(-1)^{2r+1}=(-1)\cdot(-1)^{2r}=-1$ Thus if $\displaystyle k$ is odd then the statement is true since $\displaystyle -\cos (2\pi)=-1$ hence the statement is true $\displaystyle \forall k \in \mathbb{Z}$

    3) I have to go so I will leave this one until later unless of course someone else would like to fill in the blanks.
    Last edited by Sean12345; Aug 23rd 2008 at 09:47 AM.
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  3. #3
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    I'm back.

    3) Note that $\displaystyle \sin (\theta \pm \phi) \equiv \sin (\theta) \cdot \cos (\phi) \pm \cos (\theta) \cdot \sin (\phi)$ Following from this we can see that $\displaystyle \sin (k\pi) \cdot \cos (\theta) - \cos (k\pi) \cdot sin (\theta) \equiv \sin (k\pi - \theta)$ Like previously we will split this into two seperate cases. If $\displaystyle k$ is even then $\displaystyle k=2n$ for some $\displaystyle n$ and hence $\displaystyle \sin (k\pi - \theta) = \sin (-\theta + 2n\pi)$ The sine function is also periodic with period $\displaystyle 2\pi $ and thus $\displaystyle \sin(\phi + 2x\pi) \equiv \sin (\phi)~ \forall x \in \mathbb{Z}$ Also note that $\displaystyle \sin(-\psi) \equiv -\sin (\psi)$ Knowing these two bits of information we can then see that $\displaystyle \sin (-\theta + 2n\pi) \equiv \sin (-\theta) \equiv -\sin (\theta)$ Notice that $\displaystyle (-1)^{k-1} \cdot \sin (\theta) = \frac{(-1)^{2n}}{-1} \cdot \sin (\theta) = -\sin (\theta)~\forall n \in \mathbb{Z}$ Hence the statement is true if $\displaystyle k$ is even since rather trivially $\displaystyle -\sin (\theta) = -\sin (\theta)$ .

    Now if $\displaystyle k$ is odd then $\displaystyle k=2r+1$ for some $\displaystyle r$ and thus $\displaystyle \sin (k\pi - \theta) = \sin [-\theta + (2n+1) \pi] = \sin (-\theta + 2n \pi + \pi) = \sin (-\theta + \pi)$ Note that the sine function has some of the same attributes as the cosine function in that $\displaystyle \sin (\psi +\pi) = -\sin (\psi)$ Hence using this results implies that $\displaystyle \sin (-\theta + \pi) = -\sin (-\theta) = \sin (\theta)$ Notice that $\displaystyle (-1)^{k-1} \cdot \sin (\theta) = (-1)^{2r} \cdot \sin (\theta) = \sin (\theta)~\forall r \in \mathbb {Z}$ Hence the statement is true if $\displaystyle k$ is odd since again rather trivially $\displaystyle \sin (\theta) \equiv \sin (\theta)$ and thus the statement is true $\displaystyle ~\forall k \in \mathbb {Z}$

    $\displaystyle Q.E.D. $

    Thats how the results came.
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