CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 11. Complex Numbers

Vocabulary

Trigonometric polar formof a complex number describes the location of a point on the complex plane using the
angle and the radius of the point.
The abbreviationr·cisθstands forr·(cosθ+i·sinθ)and is how trigonometric polar form is typically observed.

Guided Practice

1. Translate the following complex number from trigonometric polar form to rectangular form.
   5 cis 270◦


2. Translate the following complex number from rectangular form into trigonometric polar form.
   8


3. Multiply the following complex numbers in trigonometric polar form.
   4 cis 34◦·5 cis 16◦·^12 cis 100◦
   Answers:


4. 5 cis 270◦= 5 (cos 270◦+i·sin 270◦) = 5 ( 0 −i) =− 5 i


5. 8=8 cis 0◦


6. 
   

4 cis 34◦·5 cis 16◦·^12 cis 100◦

= 4 · 5 ·^12 ·cis( 34 ◦+ 16 ◦+ 100 ◦) =10 cis 150

Note how much easier it is to do products and quotients in trigonometric polar form.

Practice

Translate the following complex numbers from trigonometric polar form to rectangular form.

1. 5 cis 270◦


2. 2 cis 30◦
   3.−4 cisπ 4


3. 6 cisπ 3


4. 2 cis^52 π
   Translate the following complex numbers from rectangular form into trigonometric polar form.


5. 2−i


6. 5+ 12 i


7. 6i+ 8
   9.i
   Complete the following calculations and simplify.


8. 2 cis 22◦·^15 cis 15◦·3 cis 95◦


9. 9 cis 98◦÷3 cis 12◦