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M1410605Handout

M1410605Handout - SECTION 6.5 TRIG(and EULER FORMS OF A...

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SECTION 6.5: TRIG (and EULER) FORMS OF A COMPLEX NUMBER A Picture Let z = a + bi . This is plotted as a , b ( ) in the complex plane. r (modulus) The absolute value (or modulus , plural moduli ) of z is r = a + bi = a 2 + b 2 If z is a real number, then b = 0 and r = a , which is consistent with the notation for the absolute value of a real number. θ (argument) is an argument of z in the picture. (Remember that infinitely many coterminal angles can be the argument.) can be anything real if z = 0 . Finding : tan = b a (Which quadrant of the complex plane does z lie in?) (Maybe this is undefined.) Trig (or “Polar”) Form of a Complex Number z = r cos ( ) a + r sin ( ) b i or, more simply, z = r cos + i sin ( )

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Euler Form of a Complex Number Euler’s Formula: e i θ = cos + i sin Famous Case If = π , we get e i = cos + i sin 0 e i = 1 We have e something = (a negative number)!!! e i + 1 = 0 This last formula relates five of the most basic constants in mathematics: e , , i ,1, and 0 !!! From Trig Form to Euler Form z = r cos + i sin ( ) e i Trig (Polar) Form z = re i
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M1410605Handout - SECTION 6.5 TRIG(and EULER FORMS OF A...

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