The Trigonometric Form of Complex Numbers
Key Questions
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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:
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:
Now the relationship between rectangular and polar is found joining the 2 graphical representations and considering the triangle obtained:
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.
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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, replacingi^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
anglephi (usually, in radians) is defined by its trigonometric functions
sin(phi)=b/r ,
cos(phi)=a/r
(it's not defined only if botha=0 andb=0 ).
Alternatively, we can use these equations to define anglephi :
Ifa!=0 ,tan(phi)=b/a . Or, ifb!=0 ,cot(phi)=a/b . -
Trigonometric Form of Complex Numbers
z=r(cos theta + isin theta) ,where
r=|z| andtheta= Angle(z) .
I hope that this was helpful.