6tan(θ)=-2sqrt(3)

{ "query": { "display": "$$6\\tan\\left(θ\\right)=-2\\sqrt{3}$$", "symbolab_question": "EQUATION#6\\tan(θ)=-2\\sqrt{3}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{5π}{6}+πn", "degrees": "θ=150^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$6\\tan\\left(θ\\right)=-2\\sqrt{3}{\\quad:\\quad}θ=\\frac{5π}{6}+πn$$", "input": "6\\tan\\left(θ\\right)=-2\\sqrt{3}", "steps": [ { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6\\tan\\left(θ\\right)=-2\\sqrt{3}", "result": "\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6\\tan\\left(θ\\right)}{6}=\\frac{-2\\sqrt{3}}{6}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{6\\tan\\left(θ\\right)}{6}=\\frac{-2\\sqrt{3}}{6}", "result": "\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{6\\tan\\left(θ\\right)}{6}:{\\quad}\\tan\\left(θ\\right)$$", "input": "\\frac{6\\tan\\left(θ\\right)}{6}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{6}=1$$", "result": "=\\tan\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77Gr13ChTyX2aD4vx077G35oNHIXcXawymKVD6WizxD4JQJZuTAY5js+oqjdT8kslqWNgTGg+4dQ/6yu8gGhigUUqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2SruqNEn5+6b77hewXMBob4FBVzSA2xjCyDEL+lJAzwmyOP8=" } }, { "type": "interim", "title": "Simplify $$\\frac{-2\\sqrt{3}}{6}:{\\quad}-\\frac{\\sqrt{3}}{3}$$", "input": "\\frac{-2\\sqrt{3}}{6}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2\\sqrt{3}}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{\\sqrt{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7guJEacSCym+TObbGhoMBn+AfCtVUmJer3KSpKsTgy911g99dC9fj9sg0EHzBIRDRLQf313VIndcSjk9TosAOSn4APy6SRyb9uqPXtO6MrHbwt9LEn7QCBUukJKctfSJKm5KWrv5y7TlmVxX/22cMlxzRW752LLcULvsaNglcF4CWeGUAyR+d/deBRo1yUVQq" } }, { "type": "step", "result": "\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}$$", "result": "θ=\\frac{5π}{6}+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": "θ=\\frac{5π}{6}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "6\\tan(θ)+2\\sqrt{3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }