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March 28, 2017

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6. Compute the modulus and argument of each complex number. I did a-c and f.
D. -5
E. -5+5i
G. -3-4i

7. Let z= -5sqrt3/2+5/2i and w= 1+sqrt3i
a. convert z and w to polar form
b. calculate zw using De Moivres Theorem
c. calculate (z/w) using De M's theorem

Please help with these two problems? I did 1-6 and 9-10 already.

  • trig - ,

    6. I will do E, you try the others the same way

    let z = -5 + 5i
    modulus z = |z| = √((-5)^2 + 5^2) = √50 = 5√2
    argument:
    tanØ = 5/-5 = -1, where Ø is in quad II
    Ø = 135°

    G. is done the same way
    for D, think of it as -5 + 0i

  • trig - ,

    7.
    z = -5√3/2 + 5/2i
    = 5(-√3/2 + (1/2)i )
    argument :
    tan Ø = (1/2) / (-√3/2), where Ø is in II
    tan Ø = -1/√3
    Ø = 150° or 5π/6 radians

    z = 5(cos 150° + isin 150°) or 5(cos 5π/6 + isin 5π/6)

    in the same way:
    w = √10(cos 60°+ isin 60°) or √10(cos π/3 + isin π/3)

  • trig - ,

    b) to multiply two complex numbers in complex form
    if u = r1(cos Ø1 + isinØ1) and v = r2(cosØ2 + isinØ2)
    then uv = r1r2(cos(Ø1+Ø2) + isin(Ø1+Ø2)
    and u/v = r1/r2 (cos(Ø1-Ø2) + isin(Ø1-Ø2)
    so
    (z)(w) = 5√10(cos (150+60) + isin(150+60)
    = 5√10(cos 210° + isin 210°)

    do z/w using the above definition.

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