Trig. Form Of Complex Numbers:

In the complex number *a* + *b*i, where *a* is the real part and *b* is the imaginary part. ( both *a* and *b* are real numbers) The complex number can be graphed in the complex plane, which is similar to the rectangular plane, the real part are the points along the x-axis and the imaginary part are the points along the y-axis.

Complex numbers are the ordered pair (a , b).

With this, complex numbers can be treated in polar and rectangular coordinates.

Complex numbers can also be converted to vectors. Just let the complex number equal a vector variable, and the complex number can be manipulated in many different ways.

ex.

*z* = *a* + *b* i

From trig. identities:

a = r cos q

b = r sin q

z = r ( cos q + i sin q )

NOTE:

cos q + i sin q is abbreviated as cis q .

Thus, z = r cis q .

Addition of complex numbers using vectors:

ex.

s = 2 + 3i

t = -5 + 2i

s + t = (2 + 3i) + ( -5 + 2i) = -3 + 5i

Multiplication of complex numbers in trig. form:

(r cis q)(s cis j) = rs cis ( q + j )

Reciprocal of complex numbers in trig. form:

(r cis q)^{-1}= r^{-1} cis (-q)

for any integer n then;

for k = 0,1,2,3,4, . . . , n - 1