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Thread: A lot of trigonometric identities

  1. #1
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    A lot of trigonometric identities

    You should know the following trigonometric identities.
    (A)$\displaystyle sin(-x)=-sinx$
    (B)$\displaystyle cos(-x)=cosx$
    (C)$\displaystyle cos(x+y)=cosxcosy-sinxsin y$
    (D)$\displaystyle sin(x+y)=sinxcosy+cosxsin y$
    Use these to derive the following important identities, which you should also know.
    (a)$\displaystyle sin^2x+cos^2x=1$ (use C and $\displaystyle cos0=1$
    (b)$\displaystyle sin2x=2sinxcosx$
    (c)$\displaystyle cos2x=cos^2x-sin^2x$
    (d)$\displaystyle cos2x=2cos^2x-1$
    (e)$\displaystyle cos2x=1-2sin^2x$
    (f)$\displaystyle |cos\frac{x}{2}|=\sqrt{\frac{1+cosx}{2}}$
    (g)$\displaystyle |sin\frac{x}{2}|=\sqrt{\frac{1-cosx}{2}}$

    If anybody can finish these would be a tremendous help to me
    took me over 30min just to write this in latex form... sigh...
    i really hope someone will reply on how to do these with 1 or 2 as a example so i can do the rest
    Thanks in advance
    Last edited by saar4ever; Aug 24th 2009 at 06:31 PM.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by saar4ever View Post
    You should know the following trigonometric identities.
    (A)$\displaystyle sin(-x)=-sinx$
    (B)$\displaystyle cos(-x)=cosx$
    (C)$\displaystyle cos(x+y)=cosxcosy-sinxsin y$
    (D)$\displaystyle sin(x+y)=sinxcosy+cosxsin y$
    Use these to derive the following important identities, which you should also know.
    (a)$\displaystyle sin^2x+cos^2x=1$ (use C and $\displaystyle cos0=1$
    (b)$\displaystyle sin2x=2sinxcosx$
    For (a), note that if you let $\displaystyle y=-x$ in C, we have

    $\displaystyle \cos\left(x+(-x)\right)=\cos x\cos\left(-x\right)-\sin x\sin\left(-x\right)\implies \cos 0=\cos x\cos x-\sin x\left(-\sin x\right)$ $\displaystyle \implies 1=\cos^2x+\sin^2x$.

    For (b), use D and let $\displaystyle y=x$.

    Can you try the second part?
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  3. #3
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    thanks sooo much now i get how to solve these probloms just finished a b c d e now trying f and g

    thx again
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by saar4ever View Post
    You should know the following trigonometric identities.
    (A)$\displaystyle sin(-x)=-sinx$
    (B)$\displaystyle cos(-x)=cosx$
    (C)$\displaystyle cos(x+y)=cosxcosy-sinxsin y$
    (D)$\displaystyle sin(x+y)=sinxcosy+cosxsin y$
    Use these to derive the following important identities, which you should also know.

    (f)$\displaystyle |cos\frac{x}{2}|=\sqrt{\frac{1+cosx}{2}}$

    If anybody can finish these would be a tremendous help to me
    took me over 30min just to write this in latex form... sigh...
    i really hope someone will reply on how to do these with 1 or 2 as a example so i can do the rest
    Thanks in advance
    You need to use exercise (d) ($\displaystyle \cos\left(2x\right)= 2\cos^2x-1$) to solve this one.

    Note that $\displaystyle \cos\left(2x\right)=2\cos^2x-1\implies\cos^2x=\frac{\cos\left(2x\right)+1}{2}$.

    Taking the square root, we have $\displaystyle \cos x=\pm\sqrt{\frac{1+\cos\left(2x\right)}{2}}$.

    Taking the absolute value, we have $\displaystyle \left|\cos x\right|=\left|\pm\sqrt{\frac{1+\cos\left(2x\right )}{2}}\right|=\sqrt{\frac{1+\cos\left(2x\right)}{2 }}$.

    The final step is to let $\displaystyle x=\frac{y}{2}$.

    Therefore, $\displaystyle \left|\cos\left(\tfrac{y}{2}\right)\right|=\sqrt{\ frac{1+\cos\left(2\left(\tfrac{y}{2}\right)\right) }{2}}=\sqrt{\frac{1+\cos y}{2}}$.

    I hope that helps.
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  5. #5
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    thanks
    you da man
    helped me alot u are a math genius
    all i got left is g and im done thanks again
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