2cos(3θ)-1=0

{ "query": { "display": "$$2\\cos\\left(3θ\\right)-1=0$$", "symbolab_question": "EQUATION#2\\cos(3θ)-1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{π}{9}+\\frac{2πn}{3},θ=\\frac{5π}{9}+\\frac{2πn}{3}", "degrees": "θ=20^{\\circ }+120^{\\circ }n,θ=100^{\\circ }+120^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos\\left(3θ\\right)-1=0{\\quad:\\quad}θ=\\frac{π}{9}+\\frac{2πn}{3},\\:θ=\\frac{5π}{9}+\\frac{2πn}{3}$$", "input": "2\\cos\\left(3θ\\right)-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2\\cos\\left(3θ\\right)-1=0", "result": "2\\cos\\left(3θ\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "2\\cos\\left(3θ\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "2\\cos\\left(3θ\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\cos\\left(3θ\\right)=1", "result": "\\cos\\left(3θ\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\cos\\left(3θ\\right)}{2}=\\frac{1}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(3θ\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(3θ\\right)=\\frac{1}{2}$$", "result": "3θ=\\frac{π}{3}+2πn,\\:3θ=\\frac{5π}{3}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "3θ=\\frac{π}{3}+2πn,\\:3θ=\\frac{5π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$3θ=\\frac{π}{3}+2πn:{\\quad}θ=\\frac{π}{9}+\\frac{2πn}{3}$$", "input": "3θ=\\frac{π}{3}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3θ=\\frac{π}{3}+2πn", "result": "θ=\\frac{π}{9}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3θ}{3}=\\frac{\\frac{π}{3}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3θ}{3}=\\frac{\\frac{π}{3}}{3}+\\frac{2πn}{3}", "result": "θ=\\frac{π}{9}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3θ}{3}:{\\quad}θ$$", "input": "\\frac{3θ}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X3dA4MAHPdcHMed9tIcLRACWKUbvV6WK3fDUgFtg3Q+kEtLScKDV+HqoDFDPEKGuZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0gSf3Z8rinFHZ0mg86m4WQsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{3}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{π}{9}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{π}{3}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{3}}{3}=\\frac{π}{9}$$", "input": "\\frac{\\frac{π}{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{3\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=\\frac{π}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajeO2OmTiQRRyJeKIXp5SwePNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66G6jfdIOMuQQNq+PUtNxnqhu+/XUfFY44oQy7yrU6sldlcIOKMCWA2ZO2KNWsGOXEkg5cleRL9FFdzdD46/KqyFk=" } }, { "type": "step", "result": "=\\frac{π}{9}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajYtMTq+7kCE7rvSefk7TF+CddNoa0JibEfgUXb69v4MPzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhs65PJ7nXDyYj6avu1cpA3l9yPegwOu0B0iWFG/q1vMfJjcBIL5pmo83UMFZRSzJsUt8Vz4ri5I/oyZLRAY9v9S3F/y04EFlkRs6F91j6m6NPxmJzKLCps2zTe0VCJi6hO/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "θ=\\frac{π}{9}+\\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": "interim", "title": "Solve $$3θ=\\frac{5π}{3}+2πn:{\\quad}θ=\\frac{5π}{9}+\\frac{2πn}{3}$$", "input": "3θ=\\frac{5π}{3}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3θ=\\frac{5π}{3}+2πn", "result": "θ=\\frac{5π}{9}+\\frac{2πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3θ}{3}=\\frac{\\frac{5π}{3}}{3}+\\frac{2πn}{3}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{3θ}{3}=\\frac{\\frac{5π}{3}}{3}+\\frac{2πn}{3}", "result": "θ=\\frac{5π}{9}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{3θ}{3}:{\\quad}θ$$", "input": "\\frac{3θ}{3}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{3}{3}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7X3dA4MAHPdcHMed9tIcLRACWKUbvV6WK3fDUgFtg3Q+kEtLScKDV+HqoDFDPEKGuZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0gSf3Z8rinFHZ0mg86m4WQsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{5π}{3}}{3}+\\frac{2πn}{3}:{\\quad}\\frac{5π}{9}+\\frac{2πn}{3}$$", "input": "\\frac{\\frac{5π}{3}}{3}+\\frac{2πn}{3}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{5π}{3}}{3}=\\frac{5π}{9}$$", "input": "\\frac{\\frac{5π}{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5π}{3\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:3=9$$", "result": "=\\frac{5π}{9}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajeowFqk3vGjRn5MbHHAGe1rdd47a0hQ8flDbGsI5To1dYDPq1Ru8svsR9Rij2+7AACUUZ5kyTf3dT3a1QC3CdcUWtOub2dAxgaWacwKjDMkBU9pXqz+2zaB6D3OXzlyUHSLmL8RCKCKp3ERmXA9QCXMkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{5π}{9}+\\frac{2πn}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajTY6gIICShuc1OZ8SqYZP2Y2BuryoOOEqPAk8viwO4hR3XeO2tIUPH5Q2xrCOU6NXWAz6tUbvLL7EfUYo9vuwAC9kcE0uEDihVKeLKUFkaAB0xL6zoXnKidBEGB7TYSazh429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJCVlnqYRAR2jYfPV6WxDaDS+sGA/hcVpXlN2Jnq5BfzsRXZeNU0R6SHwm0bdlcOVNB" } }, { "type": "step", "result": "θ=\\frac{5π}{9}+\\frac{2πn}{3}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "2\\cos(3θ)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }