// Numbas version: exam_results_page_options
{"name": "Euler's identity", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b": {"group": "Ungrouped variables", "definition": "random(2..11#1)", "description": "", "templateType": "randrange", "name": "b"}, "a": {"group": "Ungrouped variables", "definition": "random(2..12#1)", "description": "", "templateType": "randrange", "name": "a"}, "theta": {"group": "Ungrouped variables", "definition": "arccos({c}/({a}+{b}))", "description": "", "templateType": "anything", "name": "theta"}, "d": {"group": "Ungrouped variables", "definition": "random(1..5#1)", "description": "", "templateType": "randrange", "name": "d"}, "c": {"group": "Ungrouped variables", "definition": "{a}+{b}-{d}", "description": "", "templateType": "anything", "name": "c"}, "x": {"group": "Ungrouped variables", "definition": "({a}-{b})sin({theta})", "description": "", "templateType": "anything", "name": "x"}}, "parts": [{"showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "type": "gapfill", "marks": 0, "prompt": "<p>Solve the equation for \\(\\theta\\) and \\(x\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\theta\\) =&nbsp;[[0]]</p>\n<p style=\"padding-left: 30px;\"><span color=\"#888888\" style=\"color: #888888;\">\\(x\\) =&nbsp;[[1]]</span></p>", "gaps": [{"showFeedbackIcon": true, "precision": "2", "strictPrecision": false, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "marks": 1, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "showCorrectAnswer": true, "minValue": "{theta}", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": true, "type": "numberentry", "scripts": {}, "maxValue": "{theta}"}, {"showFeedbackIcon": true, "precision": "2", "strictPrecision": false, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "marks": 1, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "variableReplacements": [], "showCorrectAnswer": true, "minValue": "{x}", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": true, "type": "numberentry", "scripts": {}, "maxValue": "{x}"}]}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "advice": "<p>\\(\\var{a}e^{j\\theta}+\\var{b}e^{-j\\theta}=\\var{c}+xj\\)</p>\n<p>\\(\\var{a}(cos(\\theta)+jsin(\\theta))+\\var{b}(cos(-\\theta)+jsin(-\\theta))=\\var{c}+xj\\)</p>\n<p>\\(\\var{a}(cos(\\theta)+jsin(\\theta))+\\var{b}(cos(\\theta)-jsin(\\theta))=\\var{c}+xj\\)</p>\n<p>\\(\\var{a}cos(\\theta)+j\\var{a}sin(\\theta))+\\var{b}cos(\\theta)-j\\var{b}sin(\\theta))=\\var{c}+xj\\)</p>\n<p>\\(\\simplify{{a}+{b}}cos(\\theta)+j\\simplify{{a}-{b}}sin(\\theta)=\\var{c}+xj\\)</p>\n<p>This equation can be split into two parts. Real = Real &nbsp;and Imaginary = Imaginary.</p>\n<p>\\(\\simplify{{a}+{b}}cos(\\theta)=\\var{c}\\) &nbsp; and &nbsp; \\(\\simplify{{a}-{b}}sin(\\theta)=x\\)</p>\n<p>\\(cos(\\theta)=\\simplify{{c}/({a}+{b})}\\) &nbsp;&nbsp;</p>\n<p>\\(\\theta = \\var{theta}\\)</p>\n<p>Inserting this into the second part gives</p>\n<p>\\(\\simplify{{a}-{b}}sin(\\var{theta})=x\\)</p>\n<p>\\(x=\\simplify{({a}-{b})}\\simplify{sin({theta})}\\)</p>\n<p>\\(x=\\var{x}\\)</p>", "extensions": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "name": "Euler's identity", "metadata": {"description": "<p>Euler's identity \\(e^{j\\theta}=cos(\\theta)+jsin(\\theta)\\)</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "b", "d", "c", "theta", "x"], "rulesets": {}, "statement": "<p>Given&nbsp;the equation \\(\\var{a}e^{j\\theta}+\\var{b}e^{-j\\theta}=\\var{c}+xj\\)</p>", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}