Trigonometry

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There are four complex fourth roots to the number 4−4√3i. These can be expressed in polar form as

z1=r1(cosθ1+isinθ1)
z2=r2(cosθ2+isinθ2)
z3=r3(cosθ3+isinθ3)
z4=r4(cosθ4+isinθ4),

where ri is a real number and 0∘≤θi<360∘. What is the value of θ1+θ2+θ3+θ4 (in degrees)?

  • Trigonometry -

    let z = 4-4√3 i
    then by De Moivre's theorem
    tanØ = -4√3/4 = -√3
    so that Ø = 120° or Ø = 300°

    z = 8(cos 120° + i sin120°) or z = 8(cos 300° + i sin300°)

    case1:
    then z^(1/4) = 8^(1/4) (cos 30° + i sin30°)
    but for tanØ = -4√3/4 , recall that the period of tanØ is 180°
    so adding 180° to our angle yields another solution
    making z(1/4) = 8^(1/4) (cos 210° + i sin 210°)
    so far we have Ø1=30° , Ø2 = 210°

    cose2:
    z^(1/4) = 8(1/2)( cos 300/4 + i sin 300/4) = 8(1/4) (cos 75 + i sin 75)
    and with a period of 180° again,
    z^(1/4) could also be 8^(1/4)(cos 255° + i sin 255°)
    giving us Ø3 = 75 and Ø4 = 255

    Ø1+Ø2+Ø3+Ø4 = 30+210+75+255 = 570°

  • Trigonometry -

    wrong man

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