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The Trigonometric Form of Complex Numbers

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Distance Between Complex Numbers in Trig Form

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Key Questions

  • Trigonometric Form of Complex Numbers

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

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


    I hope that this was helpful.

  • 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):}#

  • 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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