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\nCalculate the value of the trigonometric Fourier coefficient \\(\\frac{a_{0}}{2}\\).
\n\\(\\frac{a_{0}}{2}=\\) [[0]]
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\n\\(a_{k}\\) = [[0]]
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\n\\(b_{1}\\) =
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\n\\(b_{2}\\) = [[0]]
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\n\\(b_{n}\\) =
", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "answer": "-4*{a}/(n pi)", "customMarkingAlgorithm": "", "checkingType": "absdiff", "valuegenerators": [{"value": "", "name": "n"}], "showFeedbackIcon": true, "vsetRange": [0, 1], "customName": "", "checkingAccuracy": 0.001, "scripts": {}, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "type": "jme", "marks": "2", "showCorrectAnswer": true, "adaptiveMarkingPenalty": 0, "unitTests": [], "checkVariableNames": false}, {"prompt": "Determine an expression for the trigonometric Fourier coefficient \\(b_{n}\\), and hence evaluate \\(b_{n}\\), for n=2,4,....
\n\\(b_{n}\\) =
\n[[0]]
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", "extendBaseMarkingAlgorithm": true, "choices": ["Even", "Odd", "Neither even or odd"], "useCustomName": false, "displayType": "radiogroup", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "showCellAnswerState": true, "maxMarks": 0, "shuffleChoices": false, "customName": "", "scripts": {}, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "minMarks": 0, "type": "1_n_2", "marks": 0, "showCorrectAnswer": true, "adaptiveMarkingPenalty": 0, "unitTests": [], "distractors": ["", "", ""]}], "extensions": [], "ungrouped_variables": ["a", "k"], "statement": "Given the function:
\n\\(f(t)=\\left[\\begin{array}{cc}\\,\\,\\var{a} &\\,\\,-\\pi<t<0\\\\\\,\\,-\\var{a}&\\,\\,\\,\\,0<t<\\pi\\end{array}\\right] \\)
\n", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Calculating particular harmonic components of a Fourier series expansion.
"}, "preamble": {"css": "", "js": ""}, "advice": "\\(f(t)=\\{\\begin{array}{cc}\\,\\,\\var{a}&\\,\\,-\\pi<t<0\\\\\\,\\,-\\var{a}&\\,\\,\\,\\,\\,\\,\\,\\,\\,0<t<\\pi\\end{array}\\)
\n\\(\\frac{a_0}{2}=\\) the average value of the wave over one complete cycle
\n\\(\\frac{a_0}{2}=\\frac{\\var{a}+\\var{a}}{2}=\\simplify{({a}+{a})/2}\\)
\nAs the function is odd
\n\\(a_n=0\\)
\n\\(b_n=\\frac{1}{\\pi}\\int_{-\\pi}^{\\pi} f(t)\\sin\\left(nx\\right)dx\\)
\n\\(b_n=\\frac{1}{\\pi}\\left(\\int_{-\\pi}^{0}\\var{a}\\sin\\left({k}x\\right)dx+\\int_{0}^{\\pi}-\\var{a}\\sin\\left({n}x\\right)dx\\right)\\)
\n\\(b_n=\\frac{1}{\\pi}\\left(\\frac{\\var{2} *\\var{a} (\\cos (\\pi n) -1) }{n}\\right)\\)
\n\nIf \\(n\\) is an odd number \\(\\cos(n\\pi)=-1\\)
\n\\(b_n=\\frac{1}{\\pi}\\left(\\frac{\\var{-4} *\\var{a} }{n}\\right)\\)
\nIf \\(n\\) is an even number \\(\\cos(n\\pi)=1\\)
\n\\(b_n=0\\)
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