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{"name": "Roots of sin(z)=a", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a1"}, "v2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ln(a1-sqrt(a1^2-1)),3)", "description": "", "name": "v2"}, "v1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ln(a1+sqrt(a1^2-1)),3)", "description": "", "name": "v1"}}, "ungrouped_variables": ["a1", "v1", "v2", "tol"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Roots of sin(z)=a", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "(1+4*n)*pi/2", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "v1+tol", "minValue": "v1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "v2+tol", "minValue": "v2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$u=$ [[0]]. &nbsp;(Do not enter decimals in your answer. &nbsp;Use $n$ for the index, and if you need to enter $\\pi$, write pi.)</p>\n        <p>There are two possible numerical answers $v_1$ and $v_2$ for the imaginary part. &nbsp;Enter the&nbsp;<strong>larger</strong> value&nbsp;<strong>in the first box</strong>, and the&nbsp;<strong>smaller</strong> value&nbsp;<strong>in the second box</strong>, both to 3d.p.</p>\n        <p>$v_1=$ [[1]]. &nbsp;(Enter your answer to 3d.p.)</p>\n        <p>$v_2=$ [[2]]. &nbsp;(Enter your answer to 3d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the roots of the equation $\\sin(z)=\\var{a1}$, in the form $u+iv$, where $u$ and $v$ are real, but $u$ depends on some index $n=0,\\pm 1,\\pm 2,\\ldots$, say.</p>", "tags": ["MAS2103", "checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2012:</strong></p>\n<p>Added tags.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the roots of $\\sin(z)=a$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The equation $\\sin(z)=\\var{a1}$ can be solved by making the substitution $z=x+iy$, and using the identity</p>\n    <p>\\[\\sin(x+iy)=\\sin(x)\\cosh(y)+i\\cos(x)\\sinh(y),\\]</p>\n    <p>so then</p>\n    <p>\\[\\sin(x)\\cosh(y)+i\\cos(x)\\sinh(y)=\\var{a1}.\\]</p>\n    <p>Now equate real and imaginary parts, so</p>\n    <p>\\[\\begin{align}\\sin(x)\\cosh(y)&amp;=\\var{a1},\\tag{1}\\\\\\cos(x)\\sinh(y)&amp;=0.\\tag{2}\\end{align}\\]</p>\n    <p>From equation (2), either $\\cos(x)=0$ or $\\sinh(y)=0$.</p>\n    <p>If $\\sinh(y)=0$, then $y=0$, which would imply $\\sin(x)=\\var{a1}$ from equation (1). &nbsp;This is impossible, however, because $-1\\leqslant\\sin(x)\\leqslant 1$. &nbsp;We must have $\\cos(x)=0$, therefore, and so</p>\n    <p>\\[x=\\frac{(1+2n)\\pi}{2},\\quad n=0,\\pm 1,\\pm 2,\\ldots.\\]</p>\n    <p>Substituting this value of $x$ into equation (1) implies that $\\pm\\cosh(y)=\\var{a1}$, because $\\sin(x)=\\pm 1$.</p>\n    <p>Because $\\cosh(y)&gt;0\\;\\forall y$, however, $\\sin(x)=-1$ is not permitted. &nbsp;This occurs for $n=\\pm 1,\\pm 3,\\ldots$. &nbsp;These values of $x$ are not permitted, therefore, so we must have</p>\n    <p>\\[x=\\frac{(1+4n)\\pi}{2},\\quad n=0,\\pm 1,\\pm 2,\\ldots.\\]</p>\n    <p>then $\\cosh(y)=\\var{a1}$, and therefore</p>\n    <p>\\[y=\\operatorname{arcosh}(\\var{a1})\\;\\text{or}\\;y=-\\operatorname{arcosh}(\\var{a1}).\\]</p>\n    <p>The roots are then $u+iv$, with</p>\n    <p>\\[u=\\frac{(1+4n)\\pi}{2},\\quad n=0,\\pm 1,\\pm 2,\\ldots\\]</p>\n    <p>and</p>\n    <p>\\[v_{1,2}=\\pm\\operatorname{arcosh}(\\var{a1})=\\pm\\var{v1}\\;\\text{to 3d.p.}\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}