Ian H
Lv 7
Ian H asked in Science & MathematicsMathematics · 2 years ago

Complex angle and value question Arg (z - i)/(z + i) = pi/4 (1) It is not obvious at the start that z + i is real, so how to solve (1) ?

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  • Anonymous
    2 years ago
    Favourite answer
  • Ian H
    Lv 7
    2 years ago

    Answer supplied by Anonymous at link. Thanks. Here is my explanation of it.

    That question was about a locus. Identify all points such that

    Arg (z - i)/(z + i) = 𝛑/4

    When complex numbers are divided, their args are subtracted

    arg (z - i)/(z + i) = arg (z - i)- arg(z + i) = 𝛑/4

    arg (z - i)/(z + i) = arctan[(y – 1)/x] - arctan[(y + 1)/x] = 𝛑/4

    Using tan(A – B) = [tanA – tan B]/[1 + tanA *tan B]

    Take tangent of both sides

    [(y – 1)/x - (y + 1)/x]/[1 + (y^2 – 1)/x^2] = 1 multiply by x^2/x^2

    [xy – x – xy – x[]/[x^2 + y^2 – 1] = 1

    x^2 + y^2 – 1 = -2x

    x^2 + 2x + 1 + y^2 = 2

    (x + 1)^2 + y^2 = 2

    Circle, centre (-1, 0) radius √(2)

    Test point z = [-1 + √(7)i]/2 is on the circle. Using calculator

    arg(z – i) = 2.56821….

    arg(z + i) = 1.78281

    Their difference ~ 0.785398 ~ 𝛑/4

  • Anonymous
    2 years ago

    -arctan(2x/((x^2) + (y^2) - 1))

  • 2 years ago

    Z x -1 divided by the coefficient of 3.14 pi should do the trick

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