11.3. Trigonometric Polar Form of Complex Numbers http://www.ck12.org
r(cosθ+i·sinθ)→r·cisθ
The abbreviationr·cisθis read as “rkiss theta.” It allows you to represent a point as a radius and an angle. One
great benefit of this form is that it makes multiplying and dividing complex numbers extremely easy. For example:
Let:z 1 =r 1 ·cisθ 1 ,z 2 =r 2 ·cisθ 2 withr 26 =0.
Then:
z 1 ·z 2 =r 1 ·r 2 ·cis(θ 1 +θ 2 )
z 1 ÷z 2 =rr 21 ·cis(θ 1 −θ 2 )For basic problems, the amount of work required to compute products and quotients for complex numbers given in
either form is roughly equivalent. For more challenging questions, trigonometric polar form becomes significantly
advantageous.
Example A
Convert the following complex number from rectangular form to trigonometric polar form.
1 −
√
3 i
Solution: The radius is the absolute value of the number.
r^2 = 12 +
(
−√ 3
) 2
→r= 2The angle can be found with basic trig and the knowledge that the opposite side is always the imaginary component
and the adjacent side is always the real component.
tanθ=−
√
13 →θ=^60 ◦
Thus the trigonometric form is 2 cis 60◦.
Example B
Convert the following complex number from trigonometric polar form to rectangular form.
4 cis(^34 π)
Solution:4 cis(^34 π)= 4 (cos(^34 π)+i·sin(^34 π))= 4
(
−
√ 2
2 +
√ 2
2 i)
=− 2 √ 2 + 2 √ 2 iExample C
Divide the following complex numbers.
4 cis 32 ◦
2 cis 2 ◦
Solution:^42 ciscis^322 ◦◦=^42 cis( 32 ◦− 2 ◦) =2 cis( 30 ◦)
Concept Problem Revisited
In rectangular coordinates:
(
1 +
√
3 i)(√
2 −
√
2 i)
=
√
2 −
√
2 i+√
6 i+√
6
In trigonometric polar coordinates, 1+
√
3 i=2 cis 60◦and√
2 −
√
(^2 i=2 cis−^45 ◦. Therefore:
1 +√ 3 i
)(√
2 −
√
2 i)
=2 cis 60◦·2 cis− 45 ◦=4 cis 105◦