assignment-11-sol

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/25/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online