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Math Help - A lot of trigonometric identities

  1. #1
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    A lot of trigonometric identities

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

    \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) \implies 1=\cos^2x+\sin^2x.

    For (b), use D and let 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) sin(-x)=-sinx
    (B) cos(-x)=cosx
    (C) cos(x+y)=cosxcosy-sinxsin y
    (D) sin(x+y)=sinxcosy+cosxsin y
    Use these to derive the following important identities, which you should also know.

    (f) |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) ( \cos\left(2x\right)= 2\cos^2x-1) to solve this one.

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

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

    Taking the absolute value, we have \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 x=\frac{y}{2}.

    Therefore, \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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