The Trigonometric Form of Complex Numbers

Key Questions

  • 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.

  • Trigonometric Form of Complex Numbers

    z=r(cos theta + isin theta),

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


    I hope that this was helpful.

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