sin(2θ)=-(sqrt(3))/2

{ "query": { "display": "$$\\sin\\left(2θ\\right)=-\\frac{\\sqrt{3}}{2}$$", "symbolab_question": "EQUATION#\\sin(2θ)=-\\frac{\\sqrt{3}}{2}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{2π}{3}+πn,θ=\\frac{5π}{6}+πn", "degrees": "θ=120^{\\circ }+180^{\\circ }n,θ=150^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(2θ\\right)=-\\frac{\\sqrt{3}}{2}{\\quad:\\quad}θ=\\frac{2π}{3}+πn,\\:θ=\\frac{5π}{6}+πn$$", "input": "\\sin\\left(2θ\\right)=-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2θ\\right)=-\\frac{\\sqrt{3}}{2}$$", "result": "2θ=\\frac{4π}{3}+2πn,\\:2θ=\\frac{5π}{3}+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": "2θ=\\frac{4π}{3}+2πn,\\:2θ=\\frac{5π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2θ=\\frac{4π}{3}+2πn:{\\quad}θ=\\frac{2π}{3}+πn$$", "input": "2θ=\\frac{4π}{3}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2θ=\\frac{4π}{3}+2πn", "result": "θ=\\frac{2π}{3}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2θ}{2}=\\frac{\\frac{4π}{3}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2θ}{2}=\\frac{\\frac{4π}{3}}{2}+\\frac{2πn}{2}", "result": "θ=\\frac{2π}{3}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2θ}{2}:{\\quad}θ$$", "input": "\\frac{2θ}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7diSwzpX9ldlv2DLvSBEhWwCWKUbvV6WK3fDUgFtg3Q+kEtLScKDV+HqoDFDPEKGuZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0ksNded2fMa8XATFfdmqDhsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{4π}{3}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{2π}{3}+πn$$", "input": "\\frac{\\frac{4π}{3}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{4π}{3}}{2}=\\frac{2π}{3}$$", "input": "\\frac{\\frac{4π}{3}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{4π}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=\\frac{4π}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{2π}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajTBWoOoprKKS9IWa8nfgotbdd47a0hQ8flDbGsI5To1doX/MZ3J6bCyfsTlPrpWwV5mDmYmTRv/2useMSCAwEqsWtOub2dAxgaWacwKjDMkByEF6k2Q6xGIviNm3JNl+xGYFwmZ+SW1ReLVQ1GVGAxAkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{2π}{3}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXqIpmhnha9llx9Rfcrx9wS2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXaF/zGdyemwsn7E5T66VsFfBsJoA+PKIxZVsdKKgEm/x8LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJd8z6iaGahO7eZb2R/xzJ/GDrhX8bC74mj+aUTTaAWI7oL+85zi6ibrzOxBmpP0LLM=" } }, { "type": "step", "result": "θ=\\frac{2π}{3}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2θ=\\frac{5π}{3}+2πn:{\\quad}θ=\\frac{5π}{6}+πn$$", "input": "2θ=\\frac{5π}{3}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2θ=\\frac{5π}{3}+2πn", "result": "θ=\\frac{5π}{6}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2θ}{2}=\\frac{\\frac{5π}{3}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2θ}{2}=\\frac{\\frac{5π}{3}}{2}+\\frac{2πn}{2}", "result": "θ=\\frac{5π}{6}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2θ}{2}:{\\quad}θ$$", "input": "\\frac{2θ}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7diSwzpX9ldlv2DLvSBEhWwCWKUbvV6WK3fDUgFtg3Q+kEtLScKDV+HqoDFDPEKGuZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0ksNded2fMa8XATFfdmqDhsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{5π}{3}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{5π}{6}+πn$$", "input": "\\frac{\\frac{5π}{3}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{5π}{3}}{2}=\\frac{5π}{6}$$", "input": "\\frac{\\frac{5π}{3}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5π}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=\\frac{5π}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajSWLw7JreuHmvtqGTh/yjYXdd47a0hQ8flDbGsI5To1dYDPq1Ru8svsR9Rij2+7AAH+xs5iM2LL/4NWY+2RdqIgWtOub2dAxgaWacwKjDMkBZrJh4iAABqcmj1hIKWbzvFNJWGMLhl4OgFCpvcPsvTckt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{5π}{6}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajdeVdMzX8XcF4dIIrjmgq2u2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXWAz6tUbvLL7EfUYo9vuwAAKx1Jw4bPUEZUSv+lC2ICJ8LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJeHPmfvXfbrgKsGcF+8iHDpDrhX8bC74mj+aUTTaAWI7oL+85zi6ibrzOxBmpP0LLM=" } }, { "type": "step", "result": "θ=\\frac{5π}{6}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Fr4rkUa8A1tCdeEXjwptNifs08vu/fgWaipkNCPEGgvRnqjEAh5+DbFWcxG5WMv8002SygbK5y4m+OEMFPvH+IHEIfJSK+QubX0YlwtF7g/wyX8k9vhpXmZGSli+Og1xvwNpAil80RgW8Xvg4M5tipMOnfxgYLzy5ItQnb/dOM0Egj3Ltr2K7rldB3eo3GVwc64mKgPQ1Go3URqHT0vh/06Bk6jujmfMU/lX+i9GJFqq+Qey2Snec3Rn6G5mcMpcG9FmdvTOD15VtAVMfnSlClJjo75PkUrsXu1DRvisp7ANYuMss2ozYBJd33wIAczX++qwNZE+K+A/jMpYI1apIPc3mhkDM/CACv8KJwR2nh7/SRVJsAWT7vZTxp8YeM724IMIfYNwM2JqUy6/LsGfCeT5Rj0kETm/9hJnqz4ZzPOdiqcAK81Ko4/KpyFPiCZ8dptnCLknK03/W5+ZCJKKaQYyBJL2j075R6FHVrdPUhNadgwzuY6Jb0cogSIeKVquUcbinK14Gon5f3QBcn/JKpwBSyQ/f/TDfHNs9qNDLfIpcZPZm+pU/lbBV9einQehfhvb9mimrAoYFeuVVvHE1g==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=\\frac{2π}{3}+πn,\\:θ=\\frac{5π}{6}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\sin(2θ)+\\frac{\\sqrt{3}}{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }