test301-1-f11sol

# Test301-1-f11sol - A-AE 301 Exam 1 Fall 2011 Please put your best answer clearly written on the exam in a box Two page of notes and no calculators

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A-AE 301 Exam 1, Fall 2011. Please put your best answer clearly written on the exam in a box. Two page of notes and no calculators. NAME: 1

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Problem 1. Find the real and imaginary part for 2 i = 1 + i. To see this simply observe that 2 i = 2 ( e i π 2 ) 1 2 = 2 e i π 4 = 2 (cos( π/ 4) + i sin( π/ 4)) = 1 + i. 2
Problem 2. Compute the following ( i 1 + i ) 12 = - 1 64 To see this observe that i 12 = ( e ) 12 = e 12 = 1. Moreover, (1 + i ) 12 = ± 2 e iπ/ 4 ² 12 = 2 6 e 3 = - 64 . Hence ( i 1 + i ) 12 = i 12 (1 + i ) 12 = - 1 64 . 3

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Problem 3. Consider the equation λ 4 + 3 - 4 i = 0. Find | λ | 2 = 5 . Observe that λ 4 = - 3 + 4 i = 5 e where θ = - arctan(4 / 3). Hence | λ | 4 = | λ 4 | = | 5 e | = 5 . Therefore | λ | 2 = 5. 4
Derive an expression for the function cos( a + b ) in terms of cos( a ), cos( b ), sin( a ) and sin( b ): cos( a + b ) = cos( a ) cos( b ) - sin( a ) sin( b ) By using Euler’s formula, we have cos( a + b ) = e i ( a + b ) + e - i ( a + b ) 2 = e ia e ib + e - ia e - ib 2 = 1 2 (cos( a ) + i sin( a ))(cos( b ) + i sin( b )) + 1 2 (cos( a ) - i sin( a ))(cos( b ) - i sin( b )) = 1 2 (cos( a ) cos( b ) - sin( a ) sin( b )) + i 2 (cos( a ) sin( b ) + sin( a ) cos( b )) + 1 2 (cos( a ) cos( b ) - sin( a ) sin( b )) - i 2 (cos( a ) sin( b ) + sin( a

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## This document was uploaded on 12/29/2011.

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Test301-1-f11sol - A-AE 301 Exam 1 Fall 2011 Please put your best answer clearly written on the exam in a box Two page of notes and no calculators

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