assignment-11-sol

assignment-11-sol - MATH 135, Fall 2010 Solution of...

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MATH 135, Fall 2010 Solution of Assignment #11 Problem 1 . Prove directly that if z = re and w = se , then z w = r s e i ( θ - φ ) . Solution. z w = r e s e = r s · cos θ + i sin θ cos φ + i sin φ = r s · (cos θ + i sin θ )(cos φ - i sin φ ) (cos φ + i sin φ )(cos φ - i sin φ ) = r s · (cos θ cos φ + sin θ sin φ ) + i (sin θ cos φ - cos θ sin φ ) cos 2 φ + sin 2 φ = r s · cos( θ - φ ) + i sin( θ - φ ) 1 = r s e i ( θ - φ ) Problem 2 . Express each of the following complex numbers in standard form. (a) 4 e i 5 π/ 3 (b) (1 + i 3) 10 (c) 5 e , where θ = tan - 1 2 Solution. (a) We have 4 e i 5 π/ 3 = 4 cos 5 π 3 + i sin 5 π 3 = 4 1 2 - 3 2 i = 2 - 2 3 i (b) We have (1 + i 3) 10 = (2 e iπ/ 3 ) 10 = 2 10 e i 10 π/ 3 = 2 10 (cos 10 π 3 + i sin 10 π 3 ) = 1024( - 1 2 - 3 2 i ) = - 512 - 512 3 i. (c) We have 5 e = 5(cos θ + i sin θ ) = 5( 1 5 + 2 5 i ) = 5 + 2 5 i . Problem 3
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This note was uploaded on 07/25/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.

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assignment-11-sol - MATH 135, Fall 2010 Solution of...

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