The Trigonometric Form of Complex Numbers

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Complex Numbers in Trigonometric Form

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  • Trigonometric Form of Complex Numbers

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

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


    I hope that this was helpful.

  • Standard to polar form uses two equations - Pythagorean Theorem and the inverse tangent function.

    A complex number has two elements, the real and imaginary components. The real component is the 'x' value and the imaginary component is the 'y' value. In the same manner as vectors, you can convert the x and y components to their matching r and (theta) components.

    For example, use the complex number #3 - 4i#.

    The real component is 3 and the imaginary component is -4.

    The radius (r) is another name for the hypotenuse of a right triangle formed with these two numbers, so an application of the Pythagorean Theorem gives #sqrt(3^2+(-4)^2# and the radius is equal to 5.

    For the angle ( #theta# ), the inverse tangent function uses the ratio of y/x, or in the case of complex numbers, imaginary/real.
    #tan^(-1)((-4)/3)#
    One final step - determine the quadrant that the complex number is located within. If the real (x) is positive, and the imaginary (y) is negative, that means quadrant IV.

    If your calculator is like mine, it gives an answer of about -53.1. But if we are in quadrant IV, that means we are 53.1 degrees shy of a full 360 degree circle. The actual angle then is about 306.9 degrees. The absolute most effective way to figure out how to get the right angle is to make a quick sketch of the original complex number on an x-y graph. That way, you can immediately see which quadrant you are in.

    Quadrant I - the calculator gives the right answer
    Quadrant II and III - add 180 to the answer the calculator gives
    Quadrant IV - add 360 to the answer the calculator gives

    But regardless of these shortcuts, nothing beats making a quick sketch so you can visualize what you are working with.

    Therefore, we would express the polar form of the answer as:
    #5/_306.9^@ #

  • 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.

  • 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#.

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