Explanation Definition used: Fourier series with period 2 π : An infinite sum of sine and cosine waves with period 2 π is, f ( x ) = { a 0 + a 1 cos x + a 2 cos 2 x + ⋅ ⋅ ⋅ + a n cos n x + ⋅ ⋅ ⋅ + b 1 sin x + b 2 sin 2 x + ⋅ ⋅ ⋅ + b n sin n x + ⋅ ⋅ ⋅ } . Here, the coefficients are given by the integrals, a 0 = 1 2 π ∫ − π π f ( x ) d x , a n = 1 π ∫ − π π f ( x ) cos n x d x and b n = 1 π ∫ − π π f ( x ) sin n x d x . Calculation: The given function is f ( x ) = { 0 − π ≤ t < 0 sin t 0 ≤ t < π 2 0 π 2 < t < π . Obtain the value of a 0 as follows. a 0 = 1 2 π ∫ − π 0 0 d t + 1 2 π ∫ 0 π 2 sin t d t + 1 2 π ∫ π 2 π 0 d t = 0 − 1 2 π [ cos t ] 0 π 2 + 0 = − 1 2 π ( cos π 2 − cos 0 ) = − 1 2 π ( − 0 − 1 ) = 1 2 π Obtain the value of a n as follows. a 0 = 1 π ∫ − π 0 0 d t + 1 π ∫ 0 π 2 sin t cosn t d t + 1 π ∫ π 2 π 0 d t = 0 + 1 π ∫ 0 π 2 sin t cosn t d t + 0 = 1 π ∫ 0 π 2 sin t cosn t d t Obtain the value of a 1 by substituting n = 1 as follows. a 1 = 1 π ∫ 0 π 2 sin t cos t d t = 1 2 π ∫ 0 π 2 sin 2 t d t = − 1 4 π [ cos 2 t ] 0 π 2 = − 1 4 π ( cos 2 ( π 2 ) − cos 0 ) On further simplification, a 1 = − 1 4 π ( cos π − cos 0 ) = − 1 4 π ( − 1 − 1 ) = − 1 4 π ( − 2 ) = 1 2 π Obtain the value of a 2 by substituting n = 2 as follows. a 2 = 1 π ∫ 0 π 2 sin t cos2 t d t = 1 2 π ∫ 0 π 2 ( sin ( t + 2 t ) + sin ( t − 2 t ) ) d t = 1 2 π ∫ 0 π 2 ( sin 3 t + sin ( − t ) ) d t = 1 2 π ( − cos ( 3 t ) 3 − cos ( − t ) − 1 ) 0 π 2 It is further simplified as follows

Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780135902912

Author: Allyn J. Washington

Publisher: PEARSON

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Chapter 30, Problem 79RE

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Chapter 30, Problem 78RE

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Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

Ch. 30.1 - Prob. 1PE Ch. 30.1 - Prob. 2PE Ch. 30.1 - Prob. 1E Ch. 30.1 - Prob. 2E Ch. 30.1 - Prob. 3E Ch. 30.1 - Prob. 4E Ch. 30.1 - Prob. 5E Ch. 30.1 - Prob. 6E Ch. 30.1 - Prob. 7E Ch. 30.1 - Prob. 8E

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