sqrt(3)sec(3x)=-2

{ "query": { "display": "$$\\sqrt{3}\\sec\\left(3x\\right)=-2$$", "symbolab_question": "EQUATION#\\sqrt{3}\\sec(3x)=-2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{5π}{18}+\\frac{2πn}{3},x=\\frac{7π}{18}+\\frac{2πn}{3}", "degrees": "x=50^{\\circ }+120^{\\circ }n,x=70^{\\circ }+120^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sqrt{3}\\sec\\left(3x\\right)=-2{\\quad:\\quad}x=\\frac{5π}{18}+\\frac{2πn}{3},\\:x=\\frac{7π}{18}+\\frac{2πn}{3}$$", "input": "\\sqrt{3}\\sec\\left(3x\\right)=-2", "steps": [ { "type": "interim", "title": "Divide both sides by $$\\sqrt{3}$$", "input": "\\sqrt{3}\\sec\\left(3x\\right)=-2", "result": "\\sec\\left(3x\\right)=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\sqrt{3}$$", "result": "\\frac{\\sqrt{3}\\sec\\left(3x\\right)}{\\sqrt{3}}=\\frac{-2}{\\sqrt{3}}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{\\sqrt{3}\\sec\\left(3x\\right)}{\\sqrt{3}}=\\frac{-2}{\\sqrt{3}}", "result": "\\sec\\left(3x\\right)=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{3}\\sec\\left(3x\\right)}{\\sqrt{3}}:{\\quad}\\sec\\left(3x\\right)$$", "input": "\\frac{\\sqrt{3}\\sec\\left(3x\\right)}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$\\sqrt{3}$$", "result": "=\\sec\\left(3x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMaCT2jNH9QX+NyxMvP/M/ZBbqvHcXS1YJzMCVe3rDX+wCUCWbkwGOY7PqKo3U/JLJZS/XZurtCQMoxxNREveEE1FKk3fejFkyiOiq9iG9IkAIuSmJUDtLxRPyJ57Nkq7qib97z4ciRBzgXtcnYl8cimEMsx3N42yapE1M5oMiMwBMJa9jzdDvGquramb/qO0hA==" } }, { "type": "interim", "title": "Simplify $$\\frac{-2}{\\sqrt{3}}:{\\quad}-\\frac{2\\sqrt{3}}{3}$$", "input": "\\frac{-2}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{\\sqrt{3}}" }, { "type": "interim", "title": "Rationalize $$-\\frac{2}{\\sqrt{3}}:{\\quad}-\\frac{2\\sqrt{3}}{3}$$", "input": "-\\frac{2}{\\sqrt{3}}", "result": "=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=-\\frac{2\\sqrt{3}}{\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "interim", "title": "$$\\sqrt{3}\\sqrt{3}=3$$", "input": "\\sqrt{3}\\sqrt{3}", "result": "=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KTpmXttz9tzhFROsvdfdpl9t0rGzXfqIeCX9GJLyBFHMwViaLUXkeD+JukROhWdjqqRb86vK1jAPxwJmTBc7jxuwBfr4jqpPqXmFXXvxZ1uqe0B1E2GM2M0eTcXoyACf" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nnJxrS992qiZ26h4CpQrj1uq8dxdLVgnMwJV7esNf7AJQJZuTAY5js+oqjdT8ksl3SW91Kyp3H5B/zOUV5Tbk34APy6SRyb9uqPXtO6MrHbwt9LEn7QCBUukJKctfSJKm5KWrv5y7TlmVxX/22cMl59OGYLeLyCREU7dktZ5aMIWRUPDnIQXYMTDiaxDMpN7" } }, { "type": "step", "result": "\\sec\\left(3x\\right)=-\\frac{2\\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 $$\\sec\\left(3x\\right)=-\\frac{2\\sqrt{3}}{3}$$", "result": "3x=\\frac{5π}{6}+2πn,\\:3x=\\frac{7π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "3x=\\frac{5π}{6}+2πn,\\:3x=\\frac{7π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sec 1Eq" } }, { "type": "interim", "title": "Solve $$3x=\\frac{5π}{6}+2πn:{\\quad}x=\\frac{5π}{18}+\\frac{2πn}{3}$$", "input": "3x=\\frac{5π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=\\frac{5π}{6}+2πn", "result": "x=\\frac{5π}{18}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{\\frac{5π}{6}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=\\frac{\\frac{5π}{6}}{3}+\\frac{2πn}{3}", "result": "x=\\frac{5π}{18}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{5π}{6}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{5π}{18}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{5π}{6}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{5π}{6}}{3}=\\frac{5π}{18}$$", "input": "\\frac{\\frac{5π}{6}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5π}{6\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:3=18$$", "result": "=\\frac{5π}{18}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajQxxosvp5aVjF63+jogLTxTdd47a0hQ8flDbGsI5To1dYDPq1Ru8svsR9Rij2+7AAINFrTtmeVjw98Sxg03FGssjigtccxNWJk7pEnO8OW1Unns2iSRRLejdqfOENY6z/0gprvtbwtTy9joQ6oMf0L6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{5π}{18}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajdjgEn6+tZlZgbxRd5x1aGU2BuryoOOEqPAk8viwO4hR3XeO2tIUPH5Q2xrCOU6NXWAz6tUbvLL7EfUYo9vuwAAOMsE/qPZaawIDenVJnRRNstHGzULf3qBshJoqlnpg7U3kCh3oevUunZ7/b0qFKBSBAc1PafP4ia+acEW7bvr5XEkSXvUX801xK1X+Y0eZK2XjyEfJXvhibSbUb7rNjnESTtjBEPSGhsNB8CpKfq/H" } }, { "type": "step", "result": "x=\\frac{5π}{18}+\\frac{2πn}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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$$3$$", "input": "3x=\\frac{7π}{6}+2πn", "result": "x=\\frac{7π}{18}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{\\frac{7π}{6}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3x}{3}=\\frac{\\frac{7π}{6}}{3}+\\frac{2πn}{3}", "result": "x=\\frac{7π}{18}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3x}{3}:{\\quad}x$$", "input": "\\frac{3x}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QJV6FEydUSv+hKOzRRB7mi061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6ot8gw9eOFDGZzXCUlLc4C0" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{7π}{6}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{7π}{18}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{7π}{6}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{7π}{6}}{3}=\\frac{7π}{18}$$", "input": "\\frac{\\frac{7π}{6}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{7π}{6\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:3=18$$", "result": "=\\frac{7π}{18}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajSe/WgaJE5Kw2hGVlhAe1+vdd47a0hQ8flDbGsI5To1dVESQA48qouUS12usnKDA2INFrTtmeVjw98Sxg03FGssjigtccxNWJk7pEnO8OW1U2qLz/T5r0Kzz330r9WWMEsjO0yiEmuVETsqo70/sLrawiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{7π}{18}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajabQsPOzu+Uq7Rfprk5L1S42BuryoOOEqPAk8viwO4hR3XeO2tIUPH5Q2xrCOU6NXVREkAOPKqLlEtdrrJygwNgOMsE/qPZaawIDenVJnRRNstHGzULf3qBshJoqlnpg7U3kCh3oevUunZ7/b0qFKBSBAc1PafP4ia+acEW7bvr5B5icgSmoM4GdnAgZ3IdKjmXjyEfJXvhibSbUb7rNjnESTtjBEPSGhsNB8CpKfq/H" } }, { "type": "step", "result": "x=\\frac{7π}{18}+\\frac{2πn}{3}" } ], "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}{18}+\\frac{2πn}{3},\\:x=\\frac{7π}{18}+\\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": "\\sqrt{3}\\sec(3x)+2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }