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6.5 Trigonometric Form of a Complex number De Moivre’s theorem Graph the Complex number 2+3i On a Complex Plane imaginary real Absolute Value of the Complex number 4+3i On a Complex Plane a bi a b 2 imaginary 2 4 3i 4 3 5 2 2 real Trigonometric Form of a Complex Number Given: a + bi z r cos i sin r cos , i r sin r is called the Modulus r r a b θ is the Argument 2 b tan a 2 Writing a Complex number in Trigonometric form Point: r cos , i r sin z 4 5i Writing a Complex number in Trigonometric form Point: r cos , i r sin z 4 5i r 4 5i 4 5 41 2 2 Writing a Complex number in Trigonometric form Point: r cos , i r sin z 4 5i r 4 5i 4 5 41 2 2 128 .7 5 5 tan tan 4 4 1 51.3 so 180 51.3 128.7 51.3 Writing a Complex number in Trigonometric form Point: r cos , i r sin z 41Cos128.7 41Sin128.7 z 41Cos128.7 iSin128.7 z 4 5i r 4 5i 4 5 41 2 2 z 41 cis128.7 tan 5 5 tan 4 4 1 51.3 so 180 51.3 128.7 Product and Quotient of two Complex Numbers z r Cos iSin z r Cos iSin 1 1 1 2 2 2 1 2 z z r r Cos iSin 1 2 1 2 1 2 1 z r Cos iSin z r 1 1 1 2 2 z 0 2 2 1 2 2 Find the product z 20cis17 1 z 15cis28 2 z z 20 15cis45 1 2 300cos 45 iSin45 Find the product z 20cis17 1 z 15cis28 2 z z 20 15cis45 1 2 300cos 45 iSin45 2 2 300 i 150 2 150i 2 2 2 Find the quotient z 20cis17 1 z 15cis28 2 z z 20 15cis(17 28) 1 2 300cos(11) iSin(11) 3000.9816 (0.1908i ) 294.48 57.24i Power of Complex Numbers De Moivre’s Theorem z r Cos iSin z r Cos2 iSin2 2 2 z r Cosn iSinn n n Power of Complex Numbers Solve z 4cis135 z 16cis2 135 2 Power of Complex Numbers Solve z 4cis135 z 16cis2 135 2 z 16Cos270 iSin270 z 160 1i 16i 2 2 N th Roots of a Complex Number Where k = 0, 1, 2, …., n - 1 n 2k 2k r Cos iSin n n Find the Cube roots of 1000 K = 0, 1, 2..(3-1) done 3 2k 2k 1000 Cos iSin 3 3 20 20 10 Cos iSin 10 3 3 2(1) 21 10 Cos iSin 5 5i 3 3 3 22 22 10 Cos iSin 5 5i 3 3 3 Homework Page 457 – 459 # 1, 9, 17, 25, 33, 41, 49, 57, 65, 77, 85, 93, 101, 109 Homework Page 457 – 459 # 5, 13, 21, 29, 37, 45, 53, 61, 73, 81, 89, 97, 105, 113 “I got the Power!!!”