HW7Solution - Homework 7 Solution 1[3pt We derived a0 and ak from the following equation in the class Find bk by using T T the cosine and sine laws

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Homework 7 Solution 1. [3pt] We derived a 0 and a k from the following equation in the class. Find b k by using the cosine and sine laws and the facts that T 0 cos( αt ) dt = T 0 sin( βt ) dt = 0. Note that w 0 = 2 π/T . f ( t ) = a 0 + k =1 [ a k cos( kw 0 t ) + b k sin( kw 0 t )] Hint: Begin by multiplying ‘ sin( mw 0 t ) ’ to both sides of the equation and integrate over (0, T). Q1: Multiply sin( mw 0 t ) to both sides of the equation and integrate over(0, T): left = ± T 0 f ( t ) sin( mw 0 t ) dt right = ± T 0 a 0 sin( mw 0 t ) dt + k =1 [ a k ± T 0 cos( kw 0 t ) sin( mw 0 t ) dt + b k ± T 0 sin( kw 0 t ) sin( mw 0 ) dt ] Considering the right side The first term: ± T 0 a 0 sin( mw 0 t ) dt = 0 The second term: a k ± T 0 cos( kw 0 t ) sin( mw 0 t ) dt = 1 2 [ a k ± T 0 sin( kw 0 t + nw 0 t )+ a k ± T 0 sin( mw 0 t kw 0 t ) dt ] = 0 The third term: for any k ̸ = m b k ± T 0 sin( kw 0 t ) sin( mw 0 t ) dt = 1 2 [ b k ± T 0 cos( kw 0 t nw 0 t ) b k ± T 0 cos( mw 0 t + kw 0 t ) dt ] = 0 for k = m b k ± T
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This note was uploaded on 11/14/2011 for the course EAME 250 taught by Professor Lee during the Spring '11 term at Case Western.

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HW7Solution - Homework 7 Solution 1[3pt We derived a0 and ak from the following equation in the class Find bk by using T T the cosine and sine laws

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