cos(x-30)=2sin(x)

{ "query": { "display": "$$\\cos\\left(x-30^{\\circ\\:}\\right)=2\\sin\\left(x\\right)$$", "symbolab_question": "EQUATION#\\cos(x-30^{\\circ })=2\\sin(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=30^{\\circ }+180^{\\circ }n", "radians": "x=\\frac{π}{6}+πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(x-30^{\\circ\\:}\\right)=2\\sin\\left(x\\right){\\quad:\\quad}x=30^{\\circ\\:}+180^{\\circ\\:}n$$", "input": "\\cos\\left(x-30^{\\circ\\:}\\right)=2\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(x-30^{\\circ\\:}\\right)=2\\sin\\left(x\\right)", "result": "\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)+\\frac{1}{2}\\sin\\left(x\\right)=2\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(x-30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Use the Angle Difference identity: $$\\cos\\left(s-t\\right)=\\cos\\left(s\\right)\\cos\\left(t\\right)+\\sin\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\cos\\left(x\\right)\\cos\\left(30^{\\circ\\:}\\right)+\\sin\\left(x\\right)\\sin\\left(30^{\\circ\\:}\\right)" }, { "type": "interim", "title": "Simplify $$\\cos\\left(x\\right)\\cos\\left(30^{\\circ\\:}\\right)+\\sin\\left(x\\right)\\sin\\left(30^{\\circ\\:}\\right):{\\quad}\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)+\\frac{1}{2}\\sin\\left(x\\right)$$", "input": "\\cos\\left(x\\right)\\cos\\left(30^{\\circ\\:}\\right)+\\sin\\left(x\\right)\\sin\\left(30^{\\circ\\:}\\right)", "result": "=\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)+\\frac{1}{2}\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "Simplify $$\\cos\\left(30^{\\circ\\:}\\right):{\\quad}\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\cos\\left(30^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "secondary": [ "$$\\cos\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&180^{\\circ }&-1\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{2}&210^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{1}{2}&240^{\\circ }&-\\frac{1}{2}\\\\\\hline 90^{\\circ }&0&270^{\\circ }&0\\\\\\hline 120^{\\circ }&-\\frac{1}{2}&300^{\\circ }&\\frac{1}{2}\\\\\\hline 135^{\\circ }&-\\frac{\\sqrt{2}}{2}&315^{\\circ }&\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{2}&330^{\\circ }&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" ], "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)+\\sin\\left(30^{\\circ\\:}\\right)\\sin\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$\\sin\\left(30^{\\circ\\:}\\right):{\\quad}\\frac{1}{2}$$", "input": "\\sin\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Use the following trivial identity:$${\\quad}\\sin\\left(30^{\\circ\\:}\\right)=\\frac{1}{2}$$", "secondary": [ "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" ], "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)+\\frac{1}{2}\\sin\\left(x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbKlZG97VhsfInbP/Q5I3mu052FciCV6Q/ZuTzBHIPdDy6bFjjx5IuxvIy7/nJMTFRsLgcgrrcmkH6sZs7VXsfRCzqhIrfPcCYZJKhK1wElQ/xsYRCQ1RVKa+ODKHx99mkCO+YO8lir++vZvO9ZdRYF2CWltJ30iuIJv95P1UYiwAZEt3ZXAiqUE0HIXrrrezJJi0cdkgnw2HPjRVpcHMxZbIciYVzjaAolEaNkVONtBBRppTNgMFons9bMkTzIHooA==" } }, { "type": "step", "result": "\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)+\\frac{1}{2}\\sin\\left(x\\right)=2\\sin\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbOE48TQ0BjrZaFQS9c5sNCz6ZbkQ49+OyJsxwGcf5bHBPosKaRMq/pBPumoBrLj89/lZJuHdgx059G6Z6iwp6k/6DZLE6vHKrQJpnGEY59NT8VtHjx9MWYiGpbbdc9bCk4777sfzlA3SkctfytqklyeGNWQoq1knMYDwLuIvUGgtysIPUjHT3A7fiXubKMq8t+d78qSNOVecD9HYTQCRiLN2WVG1dxuhtwcvc5+As2ToKO+d89V0ksUsI4xf7WSo4wJYYPwif7On5mZsZiNIHrw=" } }, { "type": "step", "primary": "Subtract $$2\\sin\\left(x\\right)$$ from both sides", "result": "\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)-\\frac{3}{2}\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)-\\frac{3}{2}\\sin\\left(x\\right):{\\quad}\\frac{\\sqrt{3}\\cos\\left(x\\right)-3\\sin\\left(x\\right)}{2}$$", "input": "\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)-\\frac{3}{2}\\sin\\left(x\\right)", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)\\::{\\quad}\\frac{\\sqrt{3}\\cos\\left(x\\right)}{2}$$", "input": "\\frac{\\sqrt{3}}{2}\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{3}\\cos\\left(x\\right)}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}\\cos\\left(x\\right)}{2}-\\frac{3}{2}\\sin\\left(x\\right)" }, { "type": "interim", "title": "Multiply $$\\frac{3}{2}\\sin\\left(x\\right)\\::{\\quad}\\frac{3\\sin\\left(x\\right)}{2}$$", "input": "\\frac{3}{2}\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\sin\\left(x\\right)}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}\\cos\\left(x\\right)}{2}-\\frac{3\\sin\\left(x\\right)}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\sqrt{3}\\cos\\left(x\\right)-3\\sin\\left(x\\right)}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{\\sqrt{3}\\cos\\left(x\\right)-3\\sin\\left(x\\right)}{2}=0" }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "\\sqrt{3}\\cos\\left(x\\right)-3\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sqrt{3}\\cos\\left(x\\right)-3\\sin\\left(x\\right)=0", "result": "\\sqrt{3}-3\\tan\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\cos\\left(x\\right),\\:\\cos\\left(x\\right)\\neq0$$", "result": "\\frac{\\sqrt{3}\\cos\\left(x\\right)-3\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\frac{0}{\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Simplify", "result": "\\sqrt{3}-\\frac{3\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=0" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\tan\\left(x\\right)$$", "result": "\\sqrt{3}-3\\tan\\left(x\\right)=0" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ECl81GL+sHp+UX1O7owT7cSyLNl94gUKX42gzYJrE22qe2xrqPrVxWpZ+To2cT9q8wQyZ2Br8agbY8qkLUt6WPm0tu7qq7Iur/j8bUReEEOq5I4KKx/r65u7kTSvPypsDeZS+lPqd2qyh6wboUo5tzMXHt9Gwg4l93uDiEuYUgzwt9LEn7QCBUukJKctfSJKk2VQdqPGsmYTaYyesllHTRc4KXs8xgJNiLkJwr0b/5OJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Move $$\\sqrt{3}\\:$$to the right side", "input": "\\sqrt{3}-3\\tan\\left(x\\right)=0", "result": "-3\\tan\\left(x\\right)=-\\sqrt{3}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{3}$$ from both sides", "result": "\\sqrt{3}-3\\tan\\left(x\\right)-\\sqrt{3}=0-\\sqrt{3}" }, { "type": "step", "primary": "Simplify", "result": "-3\\tan\\left(x\\right)=-\\sqrt{3}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-3$$", "input": "-3\\tan\\left(x\\right)=-\\sqrt{3}", "result": "\\tan\\left(x\\right)=\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-3$$", "result": "\\frac{-3\\tan\\left(x\\right)}{-3}=\\frac{-\\sqrt{3}}{-3}" }, { "type": "step", "primary": "Simplify", "result": "\\tan\\left(x\\right)=\\frac{\\sqrt{3}}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\tan\\left(x\\right)=\\frac{\\sqrt{3}}{3}$$", "result": "x=30^{\\circ\\:}+180^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$180^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&1\\\\\\hline 60^{\\circ }&\\sqrt{3}\\\\\\hline 90^{\\circ }&\\pm\\infty\\\\\\hline 120^{\\circ }&-\\sqrt{3}\\\\\\hline 135^{\\circ }&-1\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=30^{\\circ\\:}+180^{\\circ\\:}n" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(x-30^{\\circ })-2\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }