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Math Help - Proving something about complex

  1. #1
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    Proving something about complex

    I need to prove that e^(it) + e^(is) = 2cos((t-s)/2)e^(i(s+t)/2). I have used lns and gotten it to -ts = i((s+t)/2)ln(2cos((t-s)/2), but I don't know where to go from there.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by dnlstffrd View Post
    I need to prove that e^(it) + e^(is) = 2cos((t-s)/2)e^(i(s+t)/2). I have used lns and gotten it to -ts = i((s+t)/2)ln(2cos((t-s)/2), but I don't know where to go from there.
    It's not that bad. Here's one way to do it. You need to know three things:

    Euler's Formula:

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

    Product-to-sum formulas for sine and cosine:

    \sin \alpha + \sin \beta = 2 \sin \frac {\alpha + \beta}{2} \cos \frac {\alpha - \beta}{2}

    and finally,

    \cos \alpha + \cos \beta = 2 \cos \frac {\alpha + \beta}{2} \cos \frac {\alpha - \beta}{2}





    Now, let's get to it:

    {\color {red}e^{it}} + {\color {blue} e^{is}} = { \color {red} \cos t + i \sin t} + { \color {blue} \cos s + i \sin s} ...........................applied Euler's formula

    = \left( \cos t + \cos s \right) + i \left( \sin t + \sin s \right) ....................................rearranged the terms

    = \left( 2 \cos \frac {s + t}{2} \cos \frac {t - s}{2} \right) + i \left( 2 \sin \frac {s + t}{2} \cos \frac {t - s}{2} \right) .........applied the sum-to-product formulas

    = 2 \cos \frac {t - s}{2} \left[ { \color {red} \cos \frac {s + t}{2} + i \sin \frac {s + t}{2}} \right] ...............................factored out the common 2 \cos \frac {t - s}{2}

    = 2 \cos \frac {t - s}{2}e^{i \frac {s + t}{2}}

    as desired
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  3. #3
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    I totally forgot about Euler's formula, once I remembered that it was a cinch lol
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  4. #4
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    I have an geometric approach to this. Try to see if you can come up with it.
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