// Numbas version: finer_feedback_settings
{"name": "Exact value cosine", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Exact value cosine", "tags": [], "metadata": {"description": "<p>Question about use of trig identities, student has to use identities to find exact value of \\(\\cos \\frac{7\\pi}{12}\\). Question is used in exam where student has to write out the solution and upload it for grading.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Show that the exact value of \\[\\cos \\frac{7\\pi}{12} = \\frac{1-\\sqrt{3} }{2\\sqrt{2}}.\\] Write your proof in your handwritten working and upload it.<br/>Hint: use the appropriate trigonometric identities and the trig ratios you know for \\(\\displaystyle{\\frac{\\pi}{4}}\\) and \\(\\displaystyle{\\frac{\\pi}{3}}\\).</p>", "advice": "<p>Using the facts that \\(\\displaystyle{\\frac\\pi{4}=\\frac{3\\pi}{12}}\\) and \\(\\displaystyle{\\frac\\pi{3}=\\frac{4\\pi}{12}}\\) we can write \\(\\displaystyle{\\frac{7\\pi}{12}=\\frac{\\pi}{4}+\\frac{\\pi}{3}}\\).</p>\n<p>Hence \\begin{align}\\cos\\frac{7\\pi}{12}&amp;=\\cos\\left(\\frac{\\pi}{4}+\\frac{\\pi}{3}\\right)\\\\&amp;=\\cos\\frac{\\pi}{4}\\cos\\frac{\\pi}{3}-\\sin\\frac{\\pi}{4}\\sin\\frac{\\pi}{3}\\\\&amp;=\\frac1{\\sqrt{2}}\\times\\frac12-\\frac1{\\sqrt{2}}\\times\\frac{\\sqrt{3}}{2}\\\\&amp;=\\frac{1-\\sqrt{3} }{2\\sqrt{2}}\\end{align}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}]}]}], "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}]}