15sin(3x)=0

{ "query": { "display": "$$15\\sin\\left(3x\\right)=0$$", "symbolab_question": "EQUATION#15\\sin(3x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{2πn}{3},x=\\frac{π}{3}+\\frac{2πn}{3}", "degrees": "x=0^{\\circ }+120^{\\circ }n,x=60^{\\circ }+120^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$15\\sin\\left(3x\\right)=0{\\quad:\\quad}x=\\frac{2πn}{3},\\:x=\\frac{π}{3}+\\frac{2πn}{3}$$", "input": "15\\sin\\left(3x\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$15$$", "input": "15\\sin\\left(3x\\right)=0", "result": "\\sin\\left(3x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$15$$", "result": "\\frac{15\\sin\\left(3x\\right)}{15}=\\frac{0}{15}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(3x\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(3x\\right)=0$$", "result": "3x=0+2πn,\\:3x=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": "3x=0+2πn,\\:3x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$3x=0+2πn:{\\quad}x=\\frac{2πn}{3}$$", "input": "3x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "3x=2πn" }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=2πn", "result": "x=\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{2πn}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{2πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$3x=π+2πn:{\\quad}x=\\frac{π}{3}+\\frac{2πn}{3}$$", "input": "3x=π+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=π+2πn", "result": "x=\\frac{π}{3}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{π}{3}+\\frac{2πn}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{3}+\\frac{2πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{2πn}{3},\\:x=\\frac{π}{3}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "15\\sin(3x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }