Transform from algebraic form to trigonometric form the complex number z

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changchengliang's profile pic

changchengliang | Elementary School Teacher | (Level 2) Adjunct Educator

Posted on

Complex numbers can be expressed in many forms:

  • Rectangular form: a + jb ,  where a and b are real numbers and j is an imaginary number given by sqrt(-1).  Here, j is used as an operator to indicate that the real number which it precedes is measured along the imaginary axis.
  • Polar form:  r /_ theta ,   where r is a positive real number which is the modulus or the magnitude of the complex number and theta is the angle of the complex number measured counter-clockwise, about the origin, from the positive real axis.  Its unit can be expressed in degrees or radians.
  • Exponential form:    r . e^j(theta) , here, the theta has to be in radians
  • Trigonometric form:   r . cos(theta) + r . j . sin(theta)

If z is a complex number, then:


= |z| /_ theta

= |z| . [ cos(theta) + j. sin(theta) ]



(112 words)


giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Since the complex number is not given, we'll write the rectangular form as:

z=x + y*i

where x is the real part,

x= Re(z) and

y is the imaginary part,

y=Im(z) and

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 = r cos t + ir sin t

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

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

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