# complex numbersTransform from algebraic form to trigonometric form the complex number z

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### 1 Answer

Rectangular form of z: z=x + y*i

x is the real part <=> x= Re(z)

y is the imaginary part <=> y=Im(z)

i is the imaginary unit, i^2=-1.

The trigonometric (polar) form is:

z=r(cos t + i sin t), where r = |z|

r = sqrt(x^2 + y^2)

r is the modulus of the complex number z.

If x and y are the coordinates of a point M(x,y), r is the position vector of the point M.

The formula for r = sqrt(x^2 + y^2) results from Pythagorean theorem, where r is hypotenuse.

cos t = x/r=>x=r cos t

sin t = y/r=>y=r sin t

z=x+iy

z = r cos t + ir sin t

tan t = y/x => t = arctan y/x

**z = r( cos t+i sin t) **- **trigonometric/polar form of the complex number z.**