2tan(x)sin(x)=sqrt(3)tan(x)

{ "query": { "display": "$$2\\tan\\left(x\\right)\\sin\\left(x\\right)=\\sqrt{3}\\tan\\left(x\\right)$$", "symbolab_question": "EQUATION#2\\tan(x)\\sin(x)=\\sqrt{3}\\tan(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=πn,x=\\frac{π}{3}+2πn,x=\\frac{2π}{3}+2πn", "degrees": "x=0^{\\circ }+180^{\\circ }n,x=60^{\\circ }+360^{\\circ }n,x=120^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\tan\\left(x\\right)\\sin\\left(x\\right)=\\sqrt{3}\\tan\\left(x\\right){\\quad:\\quad}x=πn,\\:x=\\frac{π}{3}+2πn,\\:x=\\frac{2π}{3}+2πn$$", "input": "2\\tan\\left(x\\right)\\sin\\left(x\\right)=\\sqrt{3}\\tan\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{3}\\tan\\left(x\\right)$$ from both sides", "result": "2\\tan\\left(x\\right)\\sin\\left(x\\right)-\\sqrt{3}\\tan\\left(x\\right)=0" }, { "type": "interim", "title": "Factor $$2\\tan\\left(x\\right)\\sin\\left(x\\right)-\\sqrt{3}\\tan\\left(x\\right):{\\quad}\\tan\\left(x\\right)\\left(2\\sin\\left(x\\right)-\\sqrt{3}\\right)$$", "input": "2\\tan\\left(x\\right)\\sin\\left(x\\right)-\\sqrt{3}\\tan\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$\\tan\\left(x\\right)$$", "result": "=\\tan\\left(x\\right)\\left(2\\sin\\left(x\\right)-\\sqrt{3}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\tan\\left(x\\right)\\left(2\\sin\\left(x\\right)-\\sqrt{3}\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\tan\\left(x\\right)=0\\lor\\:2\\sin\\left(x\\right)-\\sqrt{3}=0" }, { "type": "interim", "title": "$$\\tan\\left(x\\right)=0{\\quad:\\quad}x=πn$$", "input": "\\tan\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(x\\right)=0$$", "result": "x=0+πn", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=0+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+πn:{\\quad}x=πn$$", "input": "x=0+πn", "steps": [ { "type": "step", "primary": "$$0+πn=πn$$", "result": "x=πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$2\\sin\\left(x\\right)-\\sqrt{3}=0{\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{2π}{3}+2πn$$", "input": "2\\sin\\left(x\\right)-\\sqrt{3}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{3}\\:$$to the right side", "input": "2\\sin\\left(x\\right)-\\sqrt{3}=0", "result": "2\\sin\\left(x\\right)=\\sqrt{3}", "steps": [ { "type": "step", "primary": "Add $$\\sqrt{3}$$ to both sides", "result": "2\\sin\\left(x\\right)-\\sqrt{3}+\\sqrt{3}=0+\\sqrt{3}" }, { "type": "step", "primary": "Simplify", "result": "2\\sin\\left(x\\right)=\\sqrt{3}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ekOAcPFfzmZkcNOvaCOLEdwWlzgy98Z6ERQmGRkMSyCTdaV09PMxEKZ9FieghTFwXlAceOIuOPkJzb1L2shKB1RHeN8WHer5XkrY8PGocLsSUD/19Gk0rECPvht0cRZome5p+JdO9j5cRgxtA7lj30ByhvZbkK0zuz/h90ItBlvSFjMXq/GNJmgxIicJj889Bs73n/bKJk6uHxV3H84c+5q1MszLSMViXD0sbYssTyoq3crTrhe8KJ5DvAgsGF7ALB9jg3fiZAq3KTk25kiWoc8m/NUTsKS2+/nfl+eji4+/qmRbF99JTUBSmLJl1x21R33NzU6zZoE/ArHusNzdfsQVTIXxHbnuqrGIe5Uv+ELDIUjprCA+XIptxfb/EbOOB4CES9nGxD2S5ut+HDkiOxtWKyJwU+3om7eHGIbx8A80UCRHZyRKDzQfKBxnNPRTvJXrhRQTKZrV0QqJ1oeAGfve1FkLGYy7pfNHW7ge8PWtSON6otHROlKM8B29ZaSGcDUmLcVEYQyGs1xX1zlNsLEdNOgCNOpEVDz+m2v+YkGfqH9JwAAHWwoAaZnvIAzm" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\sin\\left(x\\right)=\\sqrt{3}", "result": "\\sin\\left(x\\right)=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\sin\\left(x\\right)}{2}=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(x\\right)=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=\\frac{\\sqrt{3}}{2}$$", "result": "x=\\frac{π}{3}+2πn,\\:x=\\frac{2π}{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": "x=\\frac{π}{3}+2πn,\\:x=\\frac{2π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=πn,\\:x=\\frac{π}{3}+2πn,\\:x=\\frac{2π}{3}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\tan(x)\\sin(x)-\\sqrt{3}\\tan(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }