sin(2x)=0,-pi<= x<= pi

{ "query": { "display": "$$\\sin\\left(2x\\right)=0,\\:-π\\le\\:x\\le\\:π$$", "symbolab_question": "EQUATION#\\sin(2x)=0,-π\\le x\\le π" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0,x=\\frac{π}{2},x=π,x=-π,x=-\\frac{π}{2}", "degrees": "x=0^{\\circ },x=90^{\\circ },x=180^{\\circ },x=-180^{\\circ },x=-90^{\\circ }", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(2x\\right)=0,\\:-π\\le\\:x\\le\\:π{\\quad:\\quad}x=0,\\:x=\\frac{π}{2},\\:x=π,\\:x=-π,\\:x=-\\frac{π}{2}$$", "input": "\\sin\\left(2x\\right)=0,\\:-π\\le\\:x\\le\\:π", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2x\\right)=0$$", "result": "2x=0+2πn,\\:2x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "2x=0+2πn,\\:2x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2x=0+2πn:{\\quad}x=πn$$", "input": "2x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "2x=2πn" }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=2πn", "result": "x=πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2x=π+2πn:{\\quad}x=\\frac{π}{2}+πn$$", "input": "2x=π+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=π+2πn", "result": "x=\\frac{π}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{π}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{2}+πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=πn,\\:x=\\frac{π}{2}+πn" }, { "type": "step", "primary": "Solutions for the range $$-π\\le\\:x\\le\\:π$$", "result": "x=0,\\:x=\\frac{π}{2},\\:x=π,\\:x=-π,\\:x=-\\frac{π}{2}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(2x),-π\\le x\\le π" }, "showViewLarger": true } }, "meta": { "showVerify": true } }