sin(3θ)=-1,0<= θ<= 360

{ "query": { "display": "$$\\sin\\left(3θ\\right)=-1,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}$$", "symbolab_question": "EQUATION#\\sin(3θ)=-1,0^{\\circ }\\le θ\\le 360^{\\circ }" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=90^{\\circ },θ=210^{\\circ },θ=330^{\\circ }", "radians": "θ=\\frac{π}{2},θ=\\frac{7π}{6},θ=\\frac{11π}{6}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(3θ\\right)=-1,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}{\\quad:\\quad}θ=90^{\\circ\\:},\\:θ=210^{\\circ\\:},\\:θ=330^{\\circ\\:}$$", "input": "\\sin\\left(3θ\\right)=-1,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(3θ\\right)=-1$$", "result": "3θ=270^{\\circ\\:}+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "3θ=270^{\\circ\\:}+360^{\\circ\\:}n" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$3θ=270^{\\circ\\:}+360^{\\circ\\:}n:{\\quad}θ=90^{\\circ\\:}+\\frac{360^{\\circ\\:}n}{3}$$", "input": "3θ=270^{\\circ\\:}+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3θ=270^{\\circ\\:}+360^{\\circ\\:}n", "result": "θ=90^{\\circ\\:}+\\frac{360^{\\circ\\:}n}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3θ}{3}=\\frac{270^{\\circ\\:}}{3}+\\frac{360^{\\circ\\:}n}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3θ}{3}=\\frac{270^{\\circ\\:}}{3}+\\frac{360^{\\circ\\:}n}{3}", "result": "θ=90^{\\circ\\:}+\\frac{360^{\\circ\\:}n}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3θ}{3}:{\\quad}θ$$", "input": "\\frac{3θ}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X3dA4MAHPdcHMed9tIcLRACWKUbvV6WK3fDUgFtg3Q+kEtLScKDV+HqoDFDPEKGuZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0gSf3Z8rinFHZ0mg86m4WQsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{270^{\\circ\\:}}{3}+\\frac{360^{\\circ\\:}n}{3}:{\\quad}90^{\\circ\\:}+\\frac{360^{\\circ\\:}n}{3}$$", "input": "\\frac{270^{\\circ\\:}}{3}+\\frac{360^{\\circ\\:}n}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{270^{\\circ\\:}}{3}=90^{\\circ\\:}$$", "input": "\\frac{270^{\\circ\\:}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{540^{\\circ\\:}}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=90^{\\circ\\:}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=90^{\\circ\\:}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JgEEZ1Mu+yQYAZaM3yy2K0XHvuxI1EE0DtS80PU772HNGoPE9TME3q+OPmgkv2RQ49LZFww2Cb/FDKaY0UhUnfUq+FuP5uJsPrU/ytMse308lO08eArTg2hgjFb4K57f1ZNqd0eNhA+atlGPvx/Cdp7SyhhW1YJ4JtW1aBTN448=" } }, { "type": "step", "result": "=90^{\\circ\\:}+\\frac{360^{\\circ\\:}n}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JgEEZ1Mu+yQYAZaM3yy2K+4eClzfmj7SRJMqWC1wjXrUY0DKjUNBCvXnp6XaA4I677HxSYj71RWUUXyiH0MamMzBWJotReR4P4m6RE6FZ2Py9vegdAoNT4/jvRpg8ewLfpJGz/leaNa2R3urOrGMviM7evBGD6ZNsPSHJzsN6LVN5Aod6Hr1Lp2e/29KhSgUgQHNT2nz+ImvmnBFu276+XgEvpiIclipSD8svr2M9V2rDTMB7FR6wo08yPf0kgfMQfaEjJKNRhItmdgeHD9yuA==" } }, { "type": "step", "result": "θ=90^{\\circ\\:}+\\frac{360^{\\circ\\:}n}{3}" } ], "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": "step", "result": "θ=90^{\\circ\\:}+\\frac{360^{\\circ\\:}n}{3}" }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:θ\\le\\:360^{\\circ\\:}$$", "result": "θ=90^{\\circ\\:},\\:θ=210^{\\circ\\:},\\:θ=330^{\\circ\\:}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\sin(3θ)+1,0^{\\circ }\\le θ\\le 360^{\\circ }" }, "showViewLarger": true } }, "meta": { "showVerify": true } }