tan(x/2)=sqrt(3),0<= x<2pi

{ "query": { "display": "$$\\tan\\left(\\frac{x}{2}\\right)=\\sqrt{3},\\:0\\le\\:x<2π$$", "symbolab_question": "EQUATION#\\tan(\\frac{x}{2})=\\sqrt{3},0\\le x<2π" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{2π}{3}", "degrees": "x=120^{\\circ }", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan\\left(\\frac{x}{2}\\right)=\\sqrt{3},\\:0\\le\\:x<2π{\\quad:\\quad}x=\\frac{2π}{3}$$", "input": "\\tan\\left(\\frac{x}{2}\\right)=\\sqrt{3},\\:0\\le\\:x<2π", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(\\frac{x}{2}\\right)=\\sqrt{3}$$", "result": "\\frac{x}{2}=\\frac{π}{3}+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": "\\frac{x}{2}=\\frac{π}{3}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=\\frac{π}{3}+πn:{\\quad}x=\\frac{2π}{3}+2πn$$", "input": "\\frac{x}{2}=\\frac{π}{3}+πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=\\frac{π}{3}+πn", "result": "x=\\frac{2π}{3}+2πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{3}+2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{3}+2πn", "result": "x=\\frac{2π}{3}+2πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{π}{3}+2πn:{\\quad}\\frac{2π}{3}+2πn$$", "input": "2\\cdot\\:\\frac{π}{3}+2πn", "steps": [ { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{π}{3}\\::{\\quad}\\frac{2π}{3}$$", "input": "2\\cdot\\:\\frac{π}{3}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π2}{3}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{2π}{3}+2πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvihKvBpaUh7KoJ5CHcKUNwB0gJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJ3Dp+bJB1yvfuIMe/mO4CCvvbBmbuQNTF0TphKZ8RuvatFeoIslXtlqmFYFR1OwmZBwvBbPxb16b5MJzsWVCT2PiwcvEzvOwRqXrYyQY5dyC" } }, { "type": "step", "result": "x=\\frac{2π}{3}+2πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{2π}{3}+2πn" }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:x<2π$$", "result": "x=\\frac{2π}{3}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\tan(\\frac{x}{2})-\\sqrt{3},0\\le x<2π" }, "showViewLarger": true } }, "meta": { "showVerify": true } }