The document is a study guide for precalculus chapter 9. It defines key terms related to complex numbers such as absolute value, imaginary number, modulus, and polar and rectangular forms. It provides examples of converting between polar and rectangular coordinates. It also gives examples of simplifying, adding, subtracting, multiplying, dividing and taking powers of complex numbers. The student is asked to work through these examples and express answers in rectangular or polar form depending on the question.

1. Precalculus
Chapter 9 Study Guide
Define:
Absolute Value of a Complex number
Complex Conjugates
Imaginary Number
Modulus
Polar Axis/Plane
Polar Coordinates
Polar Form of a Complex number
Rectangular Form of a Complex number
Find the rectangular coordinates of the following polar coordinates:
1. (6, 45° ) 2. (2, 330°)
æ 3p ö  π
3. ç −2, ÷
è
4.  1, 
 2
4 ø
Find the polar coordinates of the following rectangular coordinates:
5. ( − 3, - 3) 6. (5, 5)
7. (-3, 1) 8. (4, 2)
Write the equation in polar form. Round φ to the nearest degree.
9. 2x + ψ= −3 10. y = −3ξ − 4
11. 2x − 2 ψ= 6
Write each equation in rectangular form.
 π  π
12. 3 = ρχοσ θ −  13. 4 = ρχοσ θ + 
 3  2
Simplify the complex numbers.
14. i10 + ι25 15. ( 2 + 3ι) − ( 4 − 4 ι)
16. ( 2 + 7ι) + ( −3 − ι) 17. i 3 ( 4 − 3ι)
4 + 2ι
18. (i − 7 ) ( −ι + 7 ) 19.
5 − 2ι
5+ι
20.
1 − 2ι
Convert the complex number into polar form.
21. 2 + 2ι 22. −6 - 4i
23. 4 24. 3i

2. Find each product or quotient. Express the result in rectangular form.
 π π  π π
25. 4  cos + ι σ  ⋅ 3  χοσ + ι σ 
ιν ιν
 3 3  3 3
 π π  π π
26. 8  cos + ι σ  ⋅ 4  χοσ + ι σ 
ιν ιν
 4 4  2 2
 7π 7π   5π 5π 
27. 8  cos + ισιν  ÷ 2  χοσ + ι σιν 
 6 6   3 3
Find the power. Express the result in rectangular form.
8 7
28. ( 2 + 2ι) 29. ( 3 − ι)
( −1 + i ) ( −2 - 2i )
4 3
30. 31.
32. i1/ 4 33. 3
3+ι