solvefor x,2cos(x)=sqrt(3)

{ "query": { "display": "solve for $$x,\\:2\\cos\\left(x\\right)=\\sqrt{3}$$", "symbolab_question": "SOLVE_FOR#solvefor x,2\\cos(x)=\\sqrt{3}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{6}+2πn,x=\\frac{11π}{6}+2πn", "degrees": "x=30^{\\circ }+360^{\\circ }n,x=330^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:x,\\:2\\cos\\left(x\\right)=\\sqrt{3}{\\quad:\\quad}x=\\frac{π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "2\\cos\\left(x\\right)=\\sqrt{3}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\cos\\left(x\\right)=\\sqrt{3}", "result": "\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\cos\\left(x\\right)}{2}=\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=\\frac{\\sqrt{3}}{2}$$", "result": "x=\\frac{π}{6}+2πn,\\:x=\\frac{11π}{6}+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}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\cos(x)-\\sqrt{3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }