6cos(x)=2+2cos(x)

{ "query": { "display": "$$6\\cos\\left(x\\right)=2+2\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#6\\cos(x)=2+2\\cos(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{3}+2πn,x=\\frac{5π}{3}+2πn", "degrees": "x=60^{\\circ }+360^{\\circ }n,x=300^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$6\\cos\\left(x\\right)=2+2\\cos\\left(x\\right){\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "6\\cos\\left(x\\right)=2+2\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "6\\cos\\left(x\\right)=2+2\\cos\\left(x\\right)", "result": "\\cos\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "6u=2+2u" }, { "type": "interim", "title": "$$6u=2+2u{\\quad:\\quad}u=\\frac{1}{2}$$", "input": "6u=2+2u", "steps": [ { "type": "interim", "title": "Move $$2u\\:$$to the left side", "input": "6u=2+2u", "result": "4u=2", "steps": [ { "type": "step", "primary": "Subtract $$2u$$ from both sides", "result": "6u-2u=2+2u-2u" }, { "type": "step", "primary": "Simplify", "result": "4u=2" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4u=2", "result": "u=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4u}{4}=\\frac{2}{4}" }, { "type": "step", "primary": "Simplify", "result": "u=\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\frac{1}{2}{\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "\\cos\\left(x\\right)=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=\\frac{1}{2}$$", "result": "x=\\frac{π}{3}+2πn,\\:x=\\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": "x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "6\\cos(x)-2-2\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }