The Trigonometric Form of Complex Numbers

Key Questions

  • How do you convert complex numbers from standard form to polar form and vice versa?

    Answer:

    Please see the explanation below

    Explanation:

    To convert a complex number

    #z=x+iy#

    to the polar form

    #z=r(costheta+isintheta)#

    Apply the following :

    #{(r=|z|=sqrt(x^2+y^2)),(costheta=x/(|z|)),(sintheta=y/(|z|)):}#

    And to convert

    The polar form

    #z=r(costheta+isintheta)#

    to the standard form

    #z=x+iy#

    Apply the folowing

    #{(x=rcostheta),(y=rsintheta):}#

    Narad T. · 1 · Jul 13 2018
  • What is the relationship between the rectangular form of complex numbers and their corresponding polar form?

    The rectangular form of a complex form is given in terms of 2 real numbers a and b in the form: z=a+jb
    The polar form of the same number is given in terms of a magnitude r (or length) and argument q (or angle) in the form: z=r|_q
    You can "see" a complex number on a drawing in this way:
    enter image source here
    In this case the numbers a and b become the coordinates of a point representing the complex number in the special plane (Argand-Gauss) where on the x axis you plot the real part (the number a) and on the y axis the imaginary (the b number, associated with j).
    In polar form you find the same point but using the magnitude r and argument q:
    enter image source here

    Now the relationship between rectangular and polar is found joining the 2 graphical representations and considering the triangle obtained:
    enter image source here

    The relationships then are:
    1) Pitagora's Theorem (to link the length r with a and b):
    #r=sqrt(a^2+b^2)#
    2) Inverse trigonometric functions (to link the angle q with a and b):
    #q=arctan(b/a)#

    I suggest to try various complex numbers (in diferente quadrants) to see how these relationships work.

    Gió · 1 · Dec 15 2014
  • Is it possible to perform basic operations on complex numbers in polar form?

    Yes, of course.

    Polar form is very convenient to multiply complex numbers.
    Assume we have two complex numbers in polar form:
    #z_1=r_1[cos(phi_1)+i*sin(phi_1)]#
    #z_2=r_2[cos(phi_2)+i*sin(phi_2)]#
    Then their product is
    #z_1*z_2=r_1[cos(phi_1)+i*sin(phi_1)]*r_2[cos(phi_2)+i*sin(phi_2)]#
    Performing multiplication on the right, replacing #i^2# with #-1# and using trigonometric formulas for cosine and sine of a sum of two angles, we obtain
    #z_1*z_2=r_1r_2[cos(phi_1+phi_2)+i*sin(phi_1+phi_2)]#
    The above is a polar representation of a product of two complex numbers represented in polar form.

    Raising to any real power is also very convenient in polar form as this operation is an extension of multiplication:
    #{r[cos(phi)+i*sin(phi)]}^t=r^t[cos(t*phi)+i*sin(t*phi)]#

    Addition of complex numbers is much more convenient in canonical form #z=a+i*b#. That's why, to add two complex numbers in polar form, we can convert polar to canonical, add and then convert the result back to polar form.
    The first step (getting a sum in canonical form) results is
    #z_1+z_2=[r_1cos(phi_1)+r_2cos(phi_2)]+i*[r_1sin(phi_1)+r_2sin(phi_2)]#

    Converting this to a polar form can be performed according to general rule of obtaining modulus (absolute value) and argument (phase) of a complex number represented as #z=a+i*b# where
    #a=r_1cos(phi_1)+r_2cos(phi_2)# and
    #b=r_1sin(phi_1)+r_2sin(phi_2)#

    This general rule states that
    #z=r[cos(phi)+i*sin(phi)]# where
    #r=sqrt(a^2+b^2)# and
    angle #phi# (usually, in radians) is defined by its trigonometric functions
    #sin(phi)=b/r#,
    #cos(phi)=a/r#
    (it's not defined only if both #a=0# and #b=0#).
    Alternatively, we can use these equations to define angle #phi#:
    If #a!=0#, #tan(phi)=b/a#. Or, if #b!=0#, #cot(phi)=a/b#.

    Zor Shekhtman · 1 · Jan 6 2015
  • What is The Trigonometric Form of Complex Numbers?

    Trigonometric Form of Complex Numbers

    #z=r(cos theta + isin theta)#,

    where #r=|z|# and #theta=#Angle#(z)#.

    I hope that this was helpful.

    Wataru · 1 · Nov 7 2014

Questions

The Polar System