sin(2x+pi)=0

{ "query": { "display": "$$\\sin\\left(2x+π\\right)=0$$", "symbolab_question": "EQUATION#\\sin(2x+π)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=πn-\\frac{π}{2},x=πn", "degrees": "x=-90^{\\circ }+180^{\\circ }n,x=0^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(2x+π\\right)=0{\\quad:\\quad}x=πn-\\frac{π}{2},\\:x=πn$$", "input": "\\sin\\left(2x+π\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2x+π\\right)=0$$", "result": "2x+π=0+2πn,\\:2x+π=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+π=0+2πn,\\:2x+π=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2x+π=0+2πn:{\\quad}x=πn-\\frac{π}{2}$$", "input": "2x+π=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "2x+π=2πn" }, { "type": "interim", "title": "Move $$π\\:$$to the right side", "input": "2x+π=2πn", "result": "2x=2πn-π", "steps": [ { "type": "step", "primary": "Subtract $$π$$ from both sides", "result": "2x+π-π=2πn-π" }, { "type": "step", "primary": "Simplify", "result": "2x=2πn-π" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=2πn-π", "result": "x=πn-\\frac{π}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{2πn}{2}-\\frac{π}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=πn-\\frac{π}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2x+π=π+2πn:{\\quad}x=πn$$", "input": "2x+π=π+2πn", "steps": [ { "type": "step", "primary": "Subtract $$π$$ from both sides", "result": "2x+π-π=π+2πn-π" }, { "type": "step", "primary": "Simplify", "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": "step", "result": "x=πn-\\frac{π}{2},\\:x=πn" } ], "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)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }