-2sin(x)=1

{ "query": { "display": "$$-2\\sin\\left(x\\right)=1$$", "symbolab_question": "EQUATION#-2\\sin(x)=1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{7π}{6}+2πn,x=\\frac{11π}{6}+2πn", "degrees": "x=210^{\\circ }+360^{\\circ }n,x=330^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-2\\sin\\left(x\\right)=1{\\quad:\\quad}x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "-2\\sin\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2\\sin\\left(x\\right)=1", "result": "\\sin\\left(x\\right)=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2\\sin\\left(x\\right)}{-2}=\\frac{1}{-2}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(x\\right)=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-\\frac{1}{2}$$", "result": "x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+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": "x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "-2\\sin(x)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }