// Numbas version: exam_results_page_options
{"name": "Trigonometry: Solving Trigonometric Equations 1a", "extensions": ["geogebra"], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trigonometry: Solving Trigonometric Equations 1a", "tags": [], "metadata": {"description": "<p>Solving $\\sin(nx)=a$ for $x\\in (0,\\pi)$, where $n$ is an integer and $a\\in(0,1)$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve $\\sin(\\var{n}x)=\\var{a}$, for $x \\in (0,\\pi)$.&nbsp;</p>", "advice": "<p>To solve $\\sin(\\var{n}x)=\\var{a}$, we need to rearrange the equation so that it is in terms of $x$:</p>\n<p>\\[ \\begin{split}&nbsp;\\sin(\\var{n}x) &amp;\\, =\\var{a} \\\\ \\var{n}x &amp;\\,= \\sin^{-1}(\\var{a}) \\\\ x &amp;\\,= \\frac{1}{\\var{n}}\\sin^{-1}(\\var{a}). \\end{split} \\]</p>\n<p>To check how many solutions we expect to have it can be worth sketching the function and marking when it is equal to $\\var{a}$:</p>\n<p>{geogebra_applet('https://www.geogebra.org/m/pqjfb3nf',defs)}</p>\n<p>From the graph we can see that there are {solnum} solutions for $\\sin(\\var{n}x)=\\var{a}$ {solnum} between $0$ and $\\pi$.</p>\n<p>The first solution is</p>\n<p>\\[ \\begin{split} x &amp;\\,= \\frac{1}{\\var{n}}\\sin^{-1}(\\var{a}) \\\\ &amp;\\,=\\var{sol1a} \\text{ (3 d.p.)} \\end{split} \\]</p>\n<p>Since $\\sin(\\var{n}x)$ is symmetrical between between $0$ and $\\frac{\\pi}{\\var{n}}$, the second solution can be found by subtracting the first solution from $\\frac{\\pi}{\\var{n}}$:</p>\n<p>\\[ \\begin{split} x &amp;\\,= \\frac{\\pi}{\\var{n}}-\\var{sol1a} \\\\ &amp;\\,=\\var{sol2a} \\text{ (3 d.p.)}. \\end{split} \\]</p>\n<p>{advice}</p>\n<p></p>", "rulesets": {}, "extensions": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0.1..0.9 #0.1)", "description": "", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n  [\"A\",n],\n  [\"N\",a]\n]", "description": "", "templateType": "anything", "can_override": false}, "solnum": {"name": "solnum", "group": "Ungrouped variables", "definition": "if(n=2,'two','four')", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(n=2,'','{sol34}')", "description": "", "templateType": "anything", "can_override": false}, "sol34": {"name": "sol34", "group": "Ungrouped variables", "definition": "\"<p>To find the third and forth solutions, we need to add $\\\\simplify[all,fractionNumbers]{2pi/{n}}$ to each of the solutions. This is because $\\\\sin(\\\\var{n}x)$ has a period of&nbsp; $\\\\simplify[all,fractionNumbers]{2pi/{n}}$, and from the graph we can see that these points are in the second period.</p>\\n<p>Therefore, the third solution is \\\\[ \\\\begin{split} x &amp;\\\\,= \\\\var{sol1a} + \\\\simplify[all,fractionNumbers]{2pi/{n}} \\\\\\\\&nbsp; &amp;\\\\,= \\\\simplify{{sol3a}} \\\\text{ (3 d.p.)}, \\\\end{split} \\\\]</p>\\n<p>and the fourth solution is \\\\[ \\\\begin{split} x &amp;\\\\,=\\\\var{sol2a} + \\\\simplify[all,fractionNumbers]{2pi/{n}} \\\\\\\\&nbsp; &amp;\\\\,= \\\\simplify{{sol4a}}\\\\text{ (3 d.p.)}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "solution": {"name": "solution", "group": "Ungrouped variables", "definition": "if(n=2,'{solA}','{solB}')", "description": "", "templateType": "anything", "can_override": false}, "sol1": {"name": "sol1", "group": "Ungrouped variables", "definition": "(1/n)*arcsin(a)", "description": "", "templateType": "anything", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "pi/n - sol1", "description": "", "templateType": "anything", "can_override": false}, "sol3": {"name": "sol3", "group": "Ungrouped variables", "definition": "(2*pi/n)+sol1", "description": "", "templateType": "anything", "can_override": false}, "sol4": {"name": "sol4", "group": "Ungrouped variables", "definition": "2*pi/n + sol2", "description": "", "templateType": "anything", "can_override": false}, "solA": {"name": "solA", "group": "Ungrouped variables", "definition": "[sol1a,sol2a]", "description": "", "templateType": "anything", "can_override": false}, "sol1a": {"name": "sol1a", "group": "Rounded Solutions", "definition": "precround(sol1,3)", "description": "", "templateType": "anything", "can_override": false}, "sol2a": {"name": "sol2a", "group": "Rounded Solutions", "definition": "precround(sol2,3)", "description": "", "templateType": "anything", "can_override": false}, "sol3a": {"name": "sol3a", "group": "Rounded Solutions", "definition": "precround(sol3,3)", "description": "", "templateType": "anything", "can_override": false}, "sol4a": {"name": "sol4a", "group": "Rounded Solutions", "definition": "precround(sol4,3)", "description": "", "templateType": "anything", "can_override": false}, "solB": {"name": "solB", "group": "Ungrouped variables", "definition": "[sol1a, sol2a, sol3a, sol4a]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "a", "defs", "solnum", "advice", "sol34", "solution", "sol1", "sol2", "sol3", "sol4", "solA", "solB"], "variable_groups": [{"name": "Rounded Solutions", "variables": ["sol1a", "sol2a", "sol3a", "sol4a"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]</p>\n<p><em>(Give all possible solutions with your answers to 3 decimal places where necessary.)&nbsp;</em></p>", "gaps": [{"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "if({n}=2,{solA},{solB})", "allowFractions": true, "correctAnswerFractions": false, "isSet": false, "show_input_hint": true, "separator": ","}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}