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Math Help - Question on complex no 2

  1. #1
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    Question on complex no 2

    Another question, I'm really sorry.

    2.) Let z=cis(a)
    a.) Use De Moivre's theorem to show that z^n + (1/z)^n = 2cos(na)
    I was able to prove this part.
    b.) Use the binomial theorem to expand (z + (1/z))^4
    I got z^4 + 4z^2 +6 +(4/z^2) + (1/z^4), is it right?
    Part 2, Hence show that cos^4(a) = (1/8) (cos(4a) + 4cos(2a) + 3 and find the integral of cos^4(a)da.
    I'm not sure how to show..


    Thank you so so much!
    J.
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  2. #2
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    Re: Question on complex no 2

    Hey Tutu.

    The expansion is correct.

    For the last part consider the fact that e^(a*i*theta) = cos(a*theta) + i*sin(a*theta) = (e^(i*theta))^a = (cos(theta) + i*sin(theta)^a.
    Thanks from MarkFL
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  3. #3
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    Re: Question on complex no 2

    Thank you!

    But how can cos^4(a) be substituted in e^(a*i*theta) ?
    And I cannot sub it into cos(a*theta) + i*sin(a*theta) either because I will have the extra isin(a)..
    I tried it out in another way, is it that
    z^4 + 4z^2 +6 +(4/z^2) + (1/z^4) = 2cos(4a)?
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  4. #4
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    Re: Question on complex no 2

    Consider that cos is the real part and sin is the imaginary part.
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