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Math Help - Complex Numbers Basics!

  1. #1
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    Complex Numbers Basics!

    Hi I'm confused with these three questions. I do not have the answers with me so I really need your help!

    1.) If z = 4cis40, find -z.
    I know z is a+bi since 4cis40 is in the first quadrant, so -z is -a-bi, which will be in the third quadrant.
    So I got an answer of 4cis(-220) since I was told that the argument in third and fourth quadrants are always negative but the answer is 4cis220!
    Can someone help me with this?

    2.) Let Z1 = 2cis(pi/6) and z2 = rcis(delta) where r>0, and 0<=delta<2pi.
    FInd the range of values of r and delta for which z1z2 is
    a.) a real number greater than 5
    For the modulus of z1z2, I got r>5/2, which was right.
    However for the delta, I'm not sure how I'm supposed to do it, it is (pi/6) + delta >5 and then solve the inequality? Seems wrong..
    b.) a purely imaginary number with modulus less than 1
    How do I find the imaginary number?

    3.) 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: Complex Numbers Basics!

    Hey Tutu.

    The third quadrant lies in the [180,270) range which means 220 will lie in the third quadrant.

    Remember that the angle is measured in the positively orientated counter-clockwise orientation not in the negative clockwise direction.
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  3. #3
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    Re: Complex Numbers Basics!

    Hi thank you! So is it that 4cis(220) = 4cis(-140)?
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  4. #4
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    Re: Complex Numbers Basics!

    Yeah that's right.
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  5. #5
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    Re: Complex Numbers Basics!

    Thank you so much!
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