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) ?

https://www.quora.com/How-do-I-solve-arg-z-i-z+i-p...

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

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

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