ENG2086 - Engineering Mathematics 2 Solutions Q1 Question 3 a) [3 marks] [3 marks] -0.5 0 0.5 1 1.5 2 2.5 -8 -6 -4 -2 0 2 4 6 8 Even extension -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -8 -6 -4 -2 0 2 4 6 8 Odd extension b) The period is T=4sthe frequencyisHzandthe angular frequency isr/s [2] c) Fourier coefficients for the even extension: The function is even so bn=0 [1 mark] = 2 න () = 4 න 2ଶ = ்/ଶ ்/ଶ ି்/ଶ 4 4 න 2ଶ = ଶ න 2ଶ = 2ଷ 3 ቤ ଵ = 2 3 ଵ [3 marks] = 2 න () cos = න 2ଶ cos = ଵ ଶ ିଶ = 2 ቐଶ sin ฬ ଵ − 2 න sin ଵ ቑ = = 2 ൞ଶ sin ฬ ଵ − 2 ቐ − cos ฬ ଵ + 1 න cos ଵ ቑൢ = = 2 ቐଶ sin ฬ ଵ − 2 ቊ − cos ฬ ଵ + 1 sin ฬ ଵ ቋቑ = = 2 ቊ sin − 2 ൜ − cos + 1 sin ൠቋ = = 4 ൞sin 2 − 4 ቐ − cos 2 + 2sin 2 ቑൢ [9 marks] d) 1 = ସ గ ൜1 − ସ గ ቄଶ గ ቅൠ=ସ గ ቄ1 − ଼ గమ ቅ 2 = ସ ଶగ ቄ− ସ ଶగ ቅ=− ସ గమ 3 = 4 3 ቊ−1 − 4 3 ൜ −2 3ൠቋ = 4 3 ൜−1 + 8 9ଶൠ [3 marks] () = 2 + cos ஶ ୀଵ = = 1 6 + 4 ൜1 − 8 ଶൠ cos 2 − 4 ଶ cos + 4 3 ൜−1 + 8 9ଶൠ cos3 2 + ⋯ [1 mark]
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