1. 10.3A - Multiply and Divide Complex Numbers
Page 642 1)
3 units right, 2 units up 3)
8/3 units left, 5/3 units down 5)
2 units right 7)
5 units right, 6 units up 9) 13
11) 𝟑
13) 12
25)
5 (cis (0.9273))
27)
13(cis (5.1072))
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10.3A - Multiply and Divide Complex Numbers

2. 10.3A - Multiply and Divide Complex Numbers

3. 10.3 - Complex Numbers and Polar Form
Your Turn
Express 𝟏−𝒊 into polar form
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Complex Numbers and Polar Form

4. 10.3 - Complex Numbers and Polar Form
Example 7
Express 𝟐−𝟓𝒊 into polar form •
𝟐−𝟓𝒊
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Complex Numbers and Polar Form

5. 10.3 - Complex Numbers and Polar Form
Example 7
Express 𝟐−𝟓𝒊 into polar form •
𝟐−𝟓𝒊
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Complex Numbers and Polar Form

6. 10.3 - Complex Numbers and Polar Form
Example 8
Express −𝟏−𝟒𝒊 into polar form
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Complex Numbers and Polar Form

7. 10.3 - Complex Numbers and Polar Form
Your Turn
Express −𝟐+𝟓𝒊 into polar form
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Complex Numbers and Polar Form

8. Multiply and Divide Complex Numbers
Section 10.3A
Pre-Calculus AB PreAP/Dual, Revised ©2014
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10.3A - Multiply and Divide Complex Numbers

9. 10.3 - Complex Numbers and Polar Form
Review
Express 𝟑 +𝒊 into polar form •
𝟑 +𝒊
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Complex Numbers and Polar Form

10. 10.3A - Multiply and Divide Complex Numbers
Example 1
Express into standard form, 𝟐 𝐜𝐨𝐬 𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝝅 𝟔
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10.3A - Multiply and Divide Complex Numbers

11. 10.3A - Multiply and Divide Complex Numbers
Example 2
Express into standard form, 𝟖 𝐜𝐢𝐬 −𝝅 𝟑
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10.3A - Multiply and Divide Complex Numbers

12. 10.3A - Multiply and Divide Complex Numbers
Your Turn
Express into standard form, 𝟒 𝐜𝐨𝐬 𝟒𝝅 𝟑 +𝒊𝐬𝐢𝐧 𝟒𝝅 𝟑
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10.3A - Multiply and Divide Complex Numbers

13. Multiplication & Division of Complex Numbers
This rule says that to multiply two complex numbers you multiply moduli (r) and add arguments (θ), whereas to divide two complex numbers you divide moduli and subtract arguments.
𝒛 𝟏 = 𝒓 𝟏 𝐜𝐨𝐬 𝜽 𝟏 +𝒊𝐬𝐢𝐧 𝜽 𝟏 and 𝒛 𝟐 = 𝒓 𝟐 𝐜𝐨𝐬 𝜽 𝟐 +𝒊𝐬𝐢𝐧 𝜽 𝟐 are complex numbers
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10.3A - Multiply and Divide Complex Numbers

14. 10.3A - Multiply and Divide Complex Numbers
Steps
When multiplying the complex numbers, multiply the moduli (r1 and r2) and add the arguments ( 𝜽 𝟏 and 𝜽 𝟐 )
When dividing the complex numbers, divide moduli (r1 and r2) and subtract the arguments ( 𝜽 𝟏 and 𝜽 𝟐 )
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10.3A - Multiply and Divide Complex Numbers

15. 10.3 - Complex Numbers and Polar Form
Recall
Simplify (2x6)(8x3)
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Complex Numbers and Polar Form

16. 10.3 - Complex Numbers and Polar Form
Example 3
Multiply and write in standard form, 𝒛 𝟏 =𝟐 𝐜𝐨𝐬 𝟐𝝅 𝟑 +𝒊𝐬𝐢𝐧 𝟐𝝅 𝟑 and 𝒛 𝟐 =𝟖 𝐜𝐨𝐬 𝟏𝟏𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝟏𝟏𝝅 𝟔
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Complex Numbers and Polar Form

17. 10.3 - Complex Numbers and Polar Form
Example 3
Multiply and write in standard form, 𝒛 𝟏 =𝟐 𝐜𝐨𝐬 𝟐𝝅 𝟑 +𝒊𝐬𝐢𝐧 𝟐𝝅 𝟑 and 𝒛 𝟐 =𝟖 𝐜𝐨𝐬 𝟏𝟏𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝟏𝟏𝝅 𝟔
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Complex Numbers and Polar Form

18. 10.3 - Complex Numbers and Polar Form
Example 4
Write in rectangular form, 𝟐 𝐜𝐢𝐬 𝝅 𝟒 • 𝟔 𝐜𝐢𝐬 𝝅 𝟏𝟐
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Complex Numbers and Polar Form

19. 10.3 - Complex Numbers and Polar Form
Example 5
Write in standard form, 𝟐 𝐜𝐢𝐬 𝟓𝝅 𝟔 • 𝟑 𝐜𝐢𝐬 𝟕𝝅 𝟒
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Complex Numbers and Polar Form

20. 10.3 - Complex Numbers and Polar Form
Your Turn
Write in standard form, 𝟑 𝐜𝐨𝐬 𝝅 𝟑 +𝒊𝐬𝐢𝐧 𝝅 𝟑 • 𝟒 𝐜𝐨𝐬 𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝝅 𝟔
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Complex Numbers and Polar Form

21. 10.3 - Complex Numbers and Polar Form
Recall
Simplify 𝟐 𝒙 𝟔 𝟖 𝒙 𝟑
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Complex Numbers and Polar Form

22. 10.3 - Complex Numbers and Polar Form
Example 6
Divide and write in rectangular form, 𝒛 𝟏 =𝟏𝟐 𝐜𝐨𝐬 𝟓𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝟓𝝅 𝟔 𝐜𝐨𝐬 𝟓𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝟓𝝅 𝟔 and 𝒛 𝟐 =𝟑 𝐜𝐨𝐬 𝝅 𝟑 +𝒊𝐬𝐢𝐧 𝝅 𝟑 for z1/z2
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Complex Numbers and Polar Form

23. 10.3 - Complex Numbers and Polar Form
Example 6
Divide and write in rectangular form, 𝒛 𝟏 =𝟏𝟐 𝐜𝐨𝐬 𝟓𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝟓𝝅 𝟔 𝐜𝐨𝐬 𝟓𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝟓𝝅 𝟔 and 𝒛 𝟐 =𝟑 𝐜𝐨𝐬 𝝅 𝟑 +𝒊𝐬𝐢𝐧 𝝅 𝟑 for z1/z2
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Complex Numbers and Polar Form

24. 10.3 - Complex Numbers and Polar Form
Your Turn
Divide and write in rectangular form, 𝒛 𝟏 =𝟓 𝐜𝐢𝐬 𝝅 𝟒 and 𝒛 𝟐 = 𝟏 𝟑 𝐜𝐢𝐬 𝝅 𝟔 for z1/z2
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Complex Numbers and Polar Form

25. 10.3A - Multiply and Divide Complex Numbers
Assignment
Page , 27, 28, 30, odd, 39, 40, 43, 45
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10.3A - Multiply and Divide Complex Numbers