// Numbas version: exam_results_page_options
{"name": "Trig equations 3 (cos2x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trig equations 3 (cos2x)", "tags": [], "metadata": {"description": "<p>trig equation that uses cos2x=c^2-s^s</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "<p>Solve $\\simplify{2{a}{d}+2{b}{c}}sinx=\\simplify{{a}{c}cos2x-{a}{c}-2{b}{d}}$</p>\n<p>The first step is to replace $cos2x$ with $1-2sin^2x$ as below</p>\n<p>\\[ \\simplify{2{a}{d}+2{b}{c}}sinx=\\simplify{{a}{c}(1-2sin^2x)-{a}{c}-2{b}{d}} \\]</p>\n<p>If you expand the brackets and rearrange you will get the following.</p>\n<p>\\[ \\simplify{ {a}{c}sin^2x +{a}{d}sinx+{b}{c}sinx+{b}{d}} \\]</p>\n<p>which factorises to</p>\n<p>\\[ (\\simplify{{a}sinx+{b}})(\\var{c}sinx+\\var{d})=0 \\]</p>\n<p>You can now take each bracket separately</p>\n<p>\\[ sinx=\\frac{\\var{absb}}{\\var{a}} \\text{&nbsp; or&nbsp; }&nbsp; sinx=\\frac{-\\var{d}}{\\var{c}} \\]</p>\n<p>Taking each separately</p>\n<p>\\[ x=sin^{-1}\\frac{\\var{absb}}{\\var{a}} = \\var{x1} \\text{&nbsp; or&nbsp; } \\pi-\\var{x1} \\]</p>\n<p>\\[ x=sin^{-1}\\frac{-\\var{d}}{\\var{c}} = \\var{x3} \\text{&nbsp; or&nbsp; } \\pi-\\var{x3} \\]</p>\n<p>To 3 significant figures and in ascending order this gives</p>\n<p>\\[ x=\\var{x1_3} \\text{&nbsp; or&nbsp; }\\var{x2_3} \\text{&nbsp; or&nbsp; }\\var{x3_3} \\text{&nbsp; or&nbsp; }\\var{x4_3} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3,4,5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-1,-2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(3,4)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "if(b=-1,2,1)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Solutions", "definition": "arcsin(-b/a)", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Solutions", "definition": "pi-x1", "description": "", "templateType": "anything", "can_override": false}, "x3": {"name": "x3", "group": "Solutions", "definition": "pi-arcsin(-d/c)", "description": "", "templateType": "anything", "can_override": false}, "x4": {"name": "x4", "group": "Solutions", "definition": "2pi+arcsin(-d/c)", "description": "", "templateType": "anything", "can_override": false}, "absb": {"name": "absb", "group": "Ungrouped 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