tan(2x)=(-sqrt(3))/3

{ "query": { "display": "$$\\tan\\left(2x\\right)=\\frac{-\\sqrt{3}}{3}$$", "symbolab_question": "EQUATION#\\tan(2x)=\\frac{-\\sqrt{3}}{3}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{5π}{12}+\\frac{πn}{2}", "degrees": "x=75^{\\circ }+90^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan\\left(2x\\right)=\\frac{-\\sqrt{3}}{3}{\\quad:\\quad}x=\\frac{5π}{12}+\\frac{πn}{2}$$", "input": "\\tan\\left(2x\\right)=\\frac{-\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "\\tan\\left(2x\\right)=-\\frac{\\sqrt{3}}{3}" }, { "type": "interim", "title": "General solutions for $$\\tan\\left(2x\\right)=-\\frac{\\sqrt{3}}{3}$$", "result": "2x=\\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": "2x=\\frac{5π}{6}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{5π}{6}+πn:{\\quad}x=\\frac{5π}{12}+\\frac{πn}{2}$$", "input": "2x=\\frac{5π}{6}+πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{5π}{6}+πn", "result": "x=\\frac{5π}{12}+\\frac{πn}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{5π}{6}}{2}+\\frac{πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{5π}{6}}{2}+\\frac{πn}{2}", "result": "x=\\frac{5π}{12}+\\frac{πn}{2}", "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 $$\\frac{\\frac{5π}{6}}{2}+\\frac{πn}{2}:{\\quad}\\frac{5π}{12}+\\frac{πn}{2}$$", "input": "\\frac{\\frac{5π}{6}}{2}+\\frac{πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{5π}{6}}{2}=\\frac{5π}{12}$$", "input": "\\frac{\\frac{5π}{6}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5π}{6\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\frac{5π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajQHcSlptnkW86/ptUJq7cMzdd47a0hQ8flDbGsI5To1dYDPq1Ru8svsR9Rij2+7AAELhcJuE223isn9fcbq91bYjigtccxNWJk7pEnO8OW1UJKiMQdQp0z+hxSj5lYQBYQlqyL7GJZULtRYLbJSNAhCwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{5π}{12}+\\frac{πn}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZrl2Z2+wpZonfILpbPnt/+MEOV06NabtIQ9KVgA6uqjzRqDxPUzBN6vjj5oJL9kUPzGL21djq1PUuorKTcKGapmrjDphNl8OnMMfqFPHjm0joXNWqydl8x2As78T0rkuIsaY2cs9YZ/08feWmaOgkhPpaTWPScnc3H2GfYFvoQG5xgvwhuQockQAnoJLsEGBbcrZC90nLKwylUwjgxWgkKwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "x=\\frac{5π}{12}+\\frac{πn}{2}" } ], "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": "x=\\frac{5π}{12}+\\frac{πn}{2}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\tan(2x)-\\frac{-\\sqrt{3}}{3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }