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)) 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers

10.3A - Multiply and Divide Complex Numbers

10.3 - Complex Numbers and Polar Form Your Turn Express 𝟏−𝒊 into polar form 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3 - Complex Numbers and Polar Form Example 7 Express 𝟐−𝟓𝒊 into polar form • 𝟐−𝟓𝒊 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3 - Complex Numbers and Polar Form Example 7 Express 𝟐−𝟓𝒊 into polar form • 𝟐−𝟓𝒊 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3 - Complex Numbers and Polar Form Example 8 Express −𝟏−𝟒𝒊 into polar form 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3 - Complex Numbers and Polar Form Your Turn Express −𝟐+𝟓𝒊 into polar form 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

Multiply and Divide Complex Numbers Section 10.3A Pre-Calculus AB PreAP/Dual, Revised ©2014 Viet.dang@humble.k12.tx.us 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers

10.3 - Complex Numbers and Polar Form Review Express 𝟑 +𝒊 into polar form • 𝟑 +𝒊 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3A - Multiply and Divide Complex Numbers Example 1 Express into standard form, 𝟐 𝐜𝐨𝐬 𝝅 𝟔 +𝒊𝐬𝐢𝐧 𝝅 𝟔 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers

10.3A - Multiply and Divide Complex Numbers Example 2 Express into standard form, 𝟖 𝐜𝐢𝐬 −𝝅 𝟑 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers

10.3A - Multiply and Divide Complex Numbers Your Turn Express into standard form, 𝟒 𝐜𝐨𝐬 𝟒𝝅 𝟑 +𝒊𝐬𝐢𝐧 𝟒𝝅 𝟑 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers

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 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers

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 ) 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers

10.3 - Complex Numbers and Polar Form Recall Simplify (2x6)(8x3) 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

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 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

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 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3 - Complex Numbers and Polar Form Example 4 Write in rectangular form, 𝟐 𝐜𝐢𝐬 𝝅 𝟒 • 𝟔 𝐜𝐢𝐬 𝝅 𝟏𝟐 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3 - Complex Numbers and Polar Form Example 5 Write in standard form, 𝟐 𝐜𝐢𝐬 𝟓𝝅 𝟔 • 𝟑 𝐜𝐢𝐬 𝟕𝝅 𝟒 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

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 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3 - Complex Numbers and Polar Form Recall Simplify 𝟐 𝒙 𝟔 𝟖 𝒙 𝟑 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

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 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

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 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

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 11/24/2018 9:48 AM 10.3 - Complex Numbers and Polar Form

10.3A - Multiply and Divide Complex Numbers Assignment Page 642 26, 27, 28, 30, 33-37 odd, 39, 40, 43, 45 11/24/2018 9:48 AM 10.3A - Multiply and Divide Complex Numbers