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Thread: Proof of the modulus and Amplitude

  1. #1
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    Unhappy Proof of the modulus and Amplitude

    In need proof of all justifications on a moduli equation?
    x=r(cos u + i sin u)
    y=t(cos v + i sin v)
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  2. #2
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    More information please

    Hello vramirez
    Quote Originally Posted by vramirez View Post
    In need proof of all justifications on a moduli equation?
    x=r(cos u + i sin u)
    y=t(cos v + i sin v)
    Welcome to Math Help Forum!

    Sorry, but I don't understand the question. What exactly are you asked to prove?

    Grandad
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  3. #3
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    Here is my problems:

    A: Prove that the modulus of (xy) is the product of their moduli and justify my proof

    B: Prove that the amplitude of (xy) is the sum of their amplitudes and justify the proof.

    HELP!!!
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  4. #4
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    Hello vramirez
    Quote Originally Posted by vramirez View Post
    Here is my problems:

    x=r(cos u + i sin u)
    y=t(cos v + i sin v)

    A: Prove that the modulus of (xy) is the product of their moduli and justify my proof

    B: Prove that the amplitude of (xy) is the sum of their amplitudes and justify the proof.
    There are three vital things that you need to complete the proof:

    • $\displaystyle i^2 = -1$ (1)
    • $\displaystyle \cos(A+B) = \cos A\cos B -\sin A \sin B$ (2)
    • $\displaystyle \sin(A+B) = \sin A\cos B + \cos A \sin B$ (3)

    So, you simply multiply $\displaystyle x$ by $\displaystyle y$ by expanding the brackets, but leave the $\displaystyle r$ and $\displaystyle t$ outside:

    $\displaystyle xy = r(\cos u +i\sin u).t(\cos v + i\sin v)$

    $\displaystyle = rt(\cos u\cos v + \cos u.i\sin v + i\sin u \cos v + i^2\sin u \sin v)$

    $\displaystyle = rt\Big(\cos u \cos v - \sin u \sin v + i(\sin u \cos v + \cos u \sin v)\Big)$, by re-arranging the order of the terms, and using (1)

    $\displaystyle =rt\Big(\cos(u+v) + i\sin (u+v)\Big)$, using (2) and (3)

    $\displaystyle =s(\cos w + i\sin w)$, where $\displaystyle s = rt$ and $\displaystyle w = u+v$

    And this is a complex number whose modulus is $\displaystyle s$, and amplitude $\displaystyle w$, where $\displaystyle s = rt =$ product of the moduli of $\displaystyle x$ and $\displaystyle y$, and $\displaystyle w = u+v =$ sum of their amplitudes.

    Grandad
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