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Math Help - Euler's Formula

  1. #1
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    Euler's Formula

    Hello, I have a test coming up this Friday and I am completely stuck on this test review question.

    The question asks to derive the indentities using the euler's formula.

    sin2X = 2sinXcosX

    cos2X = cos^2X-sin^2X

    note: The 'X' variable stands for theta

    Any help or feedback would be greatly appreciated, thanks.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Gitano View Post
    Hello, I have a test coming up this Friday and I am completely stuck on this test review question.

    The question asks to derive the indentities using the euler's formula.

    sin2X = 2sinXcosX

    cos2X = cos^2X-sin^2X

    note: The 'X' variable stands for theta

    Any help or feedback would be greatly appreciated, thanks.
    Recall that Euler's formula is e^{i\theta}=\cos\theta+i\sin\theta. From this, we see that e^{i\theta}-e^{-i\theta}=2i\sin\theta\implies\sin\theta=\frac{e^{i  \theta}-e^{-i\theta}}{2i} and e^{i\theta}+e^{-i\theta}=2\cos\theta\implies \cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}

    Therefore,

    \begin{aligned}\sin\!\left(2\theta\right)&=\frac{e  ^{i(2\theta)}-e^{-i(2\theta)}}{2i}\\&=\frac{\left(e^{i\theta}+e^{-i\theta}\right)\left(e^{i\theta}-e^{-i\theta}\right)}{2i}\\&=2\left(\frac{e^{i\theta}+e  ^{-i\theta}}{2}\right)\left(\frac{e^{i\theta}-e^{-i\theta}}{2i}\right)\\&=2\cos\theta\sin\theta\\&=2  \sin\theta\cos\theta\end{aligned}

    Can you try to do something similar with \cos\!\left(2\theta\right)??
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  3. #3
    Behold, the power of SARDINES!
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    Quote Originally Posted by Gitano View Post
    Hello, I have a test coming up this Friday and I am completely stuck on this test review question.

    The question asks to derive the indentities using the euler's formula.

    sin2X = 2sinXcosX

    cos2X = cos^2X-sin^2X

    note: The 'X' variable stands for theta

    Any help or feedback would be greatly appreciated, thanks.
    Eulers identity is

    e^{i\theta}=\cos(\theta) + i\sin(\theta)

    (e^{i\theta})^2=e^{i2\theta}=\cos(2\theta) + i\sin(2\theta)

    but it also equals

    (e^{i\theta})^2=(\cos(\theta)+i\sin(\theta))^2=\co  s^{2}(\theta)-\sin^2(\theta)+2\sin(\theta)\cos(\theta)i

    Now setting the real and immaginary parts equal we get

    \cos(2\theta)=\cos^{2}(\theta)-\sin^2(\theta)

    \sin(2\theta)=2\sin(\theta)\cos(\theta)
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  4. #4
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    You guys are awesome, thank you.

    Although I still find it a bit confusing since euler's formula is a bit abstract, by looking at these solutions hopefully I will understand it by Friday, appreciate it.
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