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How do you find a trigonometric form of a complex number?

1 Answer

Feb 22, 2015

Let $z = x + i y$ $z=x+iy$ a complex number in algebraic form.

$z = r (cos ϕ + i sin ϕ)$ $z=r(cosphi+isinphi)$ is its trigonometric form, where:

$r = \sqrt{x^{2} + y^{2}}$ $r=sqrt(x^2+y^2)$ is the modulus of the number and

if $x > 0$ $x>0$

$ϕ = arctan (\frac{y}{x})$ $phi=arctan(y/x)$ ,

if $x < 0$ $x<0$

$ϕ = arctan (\frac{y}{x}) + π$ $phi=arctan(y/x)+pi$ ,

if $x = 0$ $x=0$ and $y > 0$ $y>0$

$ϕ = \frac{π}{2}$ $phi=pi/2$ ,

if $x = 0$ $x=0$ and $y < 0$ $y<0$

$ϕ = \frac{3}{2} π$ $phi=3/2pi$

if $x = y = 0$ $x=y=0$

It's all zero!

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