sin(30)=cos(2x)

{ "query": { "display": "$$\\sin\\left(30^{\\circ\\:}\\right)=\\cos\\left(2x\\right)$$", "symbolab_question": "EQUATION#\\sin(30^{\\circ })=\\cos(2x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=30^{\\circ }+180^{\\circ }n,x=150^{\\circ }+180^{\\circ }n", "radians": "x=\\frac{π}{6}+πn,x=\\frac{5π}{6}+πn", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(30^{\\circ\\:}\\right)=\\cos\\left(2x\\right){\\quad:\\quad}x=30^{\\circ\\:}+180^{\\circ\\:}n,\\:x=150^{\\circ\\:}+180^{\\circ\\:}n$$", "input": "\\sin\\left(30^{\\circ\\:}\\right)=\\cos\\left(2x\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(30^{\\circ\\:}\\right)=\\frac{1}{2}$$", "input": "\\sin\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(30^{\\circ\\:}\\right)=\\frac{1}{2}$$", "input": "\\sin\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Trig Evaluate" } }, { "type": "step", "result": "\\frac{1}{2}=\\cos\\left(2x\\right)" }, { "type": "step", "primary": "Switch sides", "result": "\\cos\\left(2x\\right)=\\frac{1}{2}" }, { "type": "interim", "title": "General solutions for $$\\cos\\left(2x\\right)=\\frac{1}{2}$$", "result": "2x=60^{\\circ\\:}+360^{\\circ\\:}n,\\:2x=300^{\\circ\\:}+360^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&180^{\\circ }&-1\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{2}&210^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{1}{2}&240^{\\circ }&-\\frac{1}{2}\\\\\\hline 90^{\\circ }&0&270^{\\circ }&0\\\\\\hline 120^{\\circ }&-\\frac{1}{2}&300^{\\circ }&\\frac{1}{2}\\\\\\hline 135^{\\circ }&-\\frac{\\sqrt{2}}{2}&315^{\\circ }&\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{2}&330^{\\circ }&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "2x=60^{\\circ\\:}+360^{\\circ\\:}n,\\:2x=300^{\\circ\\:}+360^{\\circ\\:}n" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$2x=60^{\\circ\\:}+360^{\\circ\\:}n:{\\quad}x=30^{\\circ\\:}+180^{\\circ\\:}n$$", "input": "2x=60^{\\circ\\:}+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=60^{\\circ\\:}+360^{\\circ\\:}n", "result": "x=30^{\\circ\\:}+180^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{60^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{60^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}", "result": "x=30^{\\circ\\:}+180^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$\\frac{60^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}:{\\quad}30^{\\circ\\:}+180^{\\circ\\:}n$$", "input": "\\frac{60^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{60^{\\circ\\:}}{2}=30^{\\circ\\:}$$", "input": "\\frac{60^{\\circ\\:}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{180^{\\circ\\:}}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=30^{\\circ\\:}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YfD7KJ+srAMCanYBrPgpXR1HTvsiFvlhwGizLG/6vzADnzlbPZjyKgy1eUCFsLd5e9nNxzEWXi43HC/7AJKGZZh89BwjZCeSZJldur0IV5fDZQeOiSyb/0BqCf5jVOmsJyNn9P5dFTzhTVSe358QTDerL0knSGi3FBYUD3Kix7A=" } }, { "type": "interim", "title": "$$\\frac{360^{\\circ\\:}n}{2}=180^{\\circ\\:}n$$", "input": "\\frac{360^{\\circ\\:}n}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=180^{\\circ\\:}n" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71VWBOBW9C7vXuDdGOZiiaITFusLFqgzJFBvrAbLH90rdd47a0hQ8flDbGsI5To1d43da852+LoWIr3J4QjOempwxvcLbE1XiLbwcsucFKH0A9deO4FPzy04NKQRXv2eP958nDhYZ5mEG7RS6wlr8Ite+cKD2yBprgOuIHg0MEWgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=30^{\\circ\\:}+180^{\\circ\\:}n" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YfD7KJ+srAMCanYBrPgpXUM8z0oIB8+sCSYYiHIRJRZmox2wkwuJl0teELkcLueqJ1UGfY00IVmqDhmhWHpTCqORWLXkjysF58uLgjK3bCtmv8mOcjoIGkEdMjARjPcDzIrJfb3/gw17D741BmoUgIEFMST8lDZxn1Yq5HMKVTtAb51BceE+a5ppiVMbLOTZ6UPFgscBZsF/YWf53CcUpk2p7iBvBavSlaiAoEtQslykcU3igz0HG4hjJBt8nPWgvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "result": "x=30^{\\circ\\:}+180^{\\circ\\:}n" } ], "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 $$2x=300^{\\circ\\:}+360^{\\circ\\:}n:{\\quad}x=150^{\\circ\\:}+180^{\\circ\\:}n$$", "input": "2x=300^{\\circ\\:}+360^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=300^{\\circ\\:}+360^{\\circ\\:}n", "result": "x=150^{\\circ\\:}+180^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{300^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{300^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}", "result": "x=150^{\\circ\\:}+180^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$\\frac{300^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}:{\\quad}150^{\\circ\\:}+180^{\\circ\\:}n$$", "input": "\\frac{300^{\\circ\\:}}{2}+\\frac{360^{\\circ\\:}n}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{300^{\\circ\\:}}{2}=150^{\\circ\\:}$$", "input": "\\frac{300^{\\circ\\:}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{900^{\\circ\\:}}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=150^{\\circ\\:}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7s3hjvpv+kWwn5NPw/LnLzibXE3Y1rwhbOmvSWpOajtjNGoPE9TME3q+OPmgkv2RQNal4MXCvbF/PgqY4sDrMWF0KrmheC67uOQV7VsC3iGfIhERcEKpL7INhO0IwQOMiT2hhbREoW+RsOwUes1lvetGHKJliOW+hSKpN1xiVZfc=" } }, { "type": "interim", "title": "$$\\frac{360^{\\circ\\:}n}{2}=180^{\\circ\\:}n$$", "input": "\\frac{360^{\\circ\\:}n}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=180^{\\circ\\:}n" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s71VWBOBW9C7vXuDdGOZiiaITFusLFqgzJFBvrAbLH90rdd47a0hQ8flDbGsI5To1d43da852+LoWIr3J4QjOempwxvcLbE1XiLbwcsucFKH0A9deO4FPzy04NKQRXv2eP958nDhYZ5mEG7RS6wlr8Ite+cKD2yBprgOuIHg0MEWgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=150^{\\circ\\:}+180^{\\circ\\:}n" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7s3hjvpv+kWwn5NPw/LnLzgzcD94eEJCE7BPiIn2prr/UY0DKjUNBCvXnp6XaA4I6nRZC8/bCRopDmEKaJ49o4czBWJotReR4P4m6RE6FZ2MT74E+rY+P42U47jZaETERioUogID0pfzOcLy5hMwYFUUqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2SruqBG3vhRDNGBfI608qB+bQilBVwQRq2eJeIvRqbXy8gJl4ksyLt5TnwYx7ZZahGZ2gjwE87HTCWyAU3ypRroDMDQ==" } }, { "type": "step", "result": "x=150^{\\circ\\:}+180^{\\circ\\:}n" } ], "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=30^{\\circ\\:}+180^{\\circ\\:}n,\\:x=150^{\\circ\\:}+180^{\\circ\\:}n" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(30^{\\circ })-\\cos(2x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }