cos(x)=cos(4x)

{ "query": { "display": "$$\\cos\\left(x\\right)=\\cos\\left(4x\\right)$$", "symbolab_question": "EQUATION#\\cos(x)=\\cos(4x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{4πn}{3},x=\\frac{2π}{3}+\\frac{4πn}{3},x=\\frac{4πn}{5},x=\\frac{2π}{5}+\\frac{4πn}{5}", "degrees": "x=0^{\\circ }+240^{\\circ }n,x=120^{\\circ }+240^{\\circ }n,x=0^{\\circ }+144^{\\circ }n,x=72^{\\circ }+144^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\cos\\left(4x\\right){\\quad:\\quad}x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3},\\:x=\\frac{4πn}{5},\\:x=\\frac{2π}{5}+\\frac{4πn}{5}$$", "input": "\\cos\\left(x\\right)=\\cos\\left(4x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cos\\left(4x\\right)$$ from both sides", "result": "\\cos\\left(x\\right)-\\cos\\left(4x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-\\cos\\left(4x\\right)+\\cos\\left(x\\right)", "result": "2\\sin\\left(\\frac{3x}{2}\\right)\\sin\\left(\\frac{5x}{2}\\right)=0", "steps": [ { "type": "step", "primary": "Use the Sum to Product identity: $$\\cos\\left(s\\right)-\\cos\\left(t\\right)=-2\\sin\\left(\\frac{s+t}{2}\\right)\\sin\\left(\\frac{s-t}{2}\\right)$$", "result": "=-2\\sin\\left(\\frac{x+4x}{2}\\right)\\sin\\left(\\frac{x-4x}{2}\\right)" }, { "type": "interim", "title": "Simplify $$-2\\sin\\left(\\frac{x+4x}{2}\\right)\\sin\\left(\\frac{x-4x}{2}\\right):{\\quad}2\\sin\\left(\\frac{3x}{2}\\right)\\sin\\left(\\frac{5x}{2}\\right)$$", "input": "-2\\sin\\left(\\frac{x+4x}{2}\\right)\\sin\\left(\\frac{x-4x}{2}\\right)", "result": "=2\\sin\\left(\\frac{3x}{2}\\right)\\sin\\left(\\frac{5x}{2}\\right)", "steps": [ { "type": "step", "primary": "Add similar elements: $$x+4x=5x$$", "result": "=-2\\sin\\left(\\frac{5x}{2}\\right)\\sin\\left(\\frac{x-4x}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{x-4x}{2}=-\\frac{3x}{2}$$", "input": "\\frac{x-4x}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$x-4x=-3x$$", "result": "=\\frac{-3x}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{3x}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hJuMvzmUGBl4XGbcZLjWrHyRHuGw7+tM5METTDj6vVHDFYNeJ0RX0LOwY+dS7jXMWfXFe5iLekp59T95n7bOdAeTV4u20zW11V5PLOwL6WsmGCGq9UAuxMbABs1r+rl1h3tSSDRgUVIGIiVhjAVCBQ==" } }, { "type": "step", "result": "=-2\\sin\\left(\\frac{5x}{2}\\right)\\sin\\left(-\\frac{3x}{2}\\right)" }, { "type": "step", "primary": "Use the negative angle identity: $$\\sin\\left(-x\\right)=-\\sin\\left(x\\right)$$", "secondary": [], "result": "=-2\\left(-\\sin\\left(\\frac{3x}{2}\\right)\\right)\\sin\\left(\\frac{5x}{2}\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=2\\sin\\left(\\frac{3x}{2}\\right)\\sin\\left(\\frac{5x}{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WKQDtlNbtfOFijcUoWrkOm11aMD22p08RlIhG/ycPOcSKEQ/GAOPLLXl5rye5qN+AJYpRu9XpYrd8NSAW2DdD1qcvkGoOmj7vUKbPyNbZzFru2QMWWcfoLh0hqhESZYd+OIV2x0BxhoBlZj/5JoUGL8/aaV6hoEMlzbAaLUjcF8eNvb7k0sVmuwf19w9aD9N8ICFoV4bcDmJnA5qsnrFkPh5GrWC+L43hpkiUxdogTcHjLyLq26xsvc5O6aMCsvWuy3AjrVrKqaqUbsR5/pakA==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sxSOfOl7Um+zqSbarNITtt/Ql2UZdOMXJCii+16T/owFMS0G0HgjUXwc3RQOmn9BYznLdIBaHD4ibBnEILJTDXUHW1mE5UQUTIwuQd6ag+xNQUndUJlCvVvSnNUC8d6va7tkDFlnH6C4dIaoREmWHfjiFdsdAcYaAZWY/+SaFBgUKDL7xbphEshRUIficWI8q73txO3A5cA/BvVWH7unbk2Rh/Eqh7VFrmGtnacNG7pYbHfPUw87gtwVLoPj4+3k" } }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(\\frac{3x}{2}\\right)=0\\lor\\:\\sin\\left(\\frac{5x}{2}\\right)=0" }, { "type": "interim", "title": "$$\\sin\\left(\\frac{3x}{2}\\right)=0{\\quad:\\quad}x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3}$$", "input": "\\sin\\left(\\frac{3x}{2}\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(\\frac{3x}{2}\\right)=0$$", "result": "\\frac{3x}{2}=0+2πn,\\:\\frac{3x}{2}=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "\\frac{3x}{2}=0+2πn,\\:\\frac{3x}{2}=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{3x}{2}=0+2πn:{\\quad}x=\\frac{4πn}{3}$$", "input": "\\frac{3x}{2}=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "\\frac{3x}{2}=2πn" }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{3x}{2}=2πn", "result": "3x=4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:3x}{2}=2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "3x=4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=4πn", "result": "x=\\frac{4πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{4πn}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{4πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72xtvzwvUh2TrjE07cWTZddGeqMQCHn4NsVZzEblYy/zTTZLKBsrnLib44QwU+8f4aUr+L090mJMD3/HR2m05UqbuFRVQd47vlEnyhWh+9b42Reueoo4ZWM0cwPZf/fZ9ic2+FidoCLUBtPd5P/cqkJ9AzlAioTbPkbJdA2+8wLrb/gv4KuDq/FafanOKSskCe9yYF2Q5qSv3gDlLQoBFjrCFOmmgubeaxj8+f0tfQPEAPg8V+ESA8anNKjrX9jbFDT1FcZViXVWPxaIkmQT8JRHz/0XLf9riUfIdtZxaSThsdLk8wNLursBNgrE2E6YPfZJsk0bZPOppU/zx6xcnfkd9zc1Os2aBPwKx7rDc3X7EFUyF8R257qqxiHuVL/hCwyFI6awgPlyKbcX2/xGzjgeAhEvZxsQ9kubrfhw5IjsbVisicFPt6Ju3hxiG8fAPNFAkR2ckSg80HygcZzT0U7yV64UUEyma1dEKidaHgBn73tRZCxmMu6XzR1u4HvD1rUjrJZ+WwWW3dvO1wqNupzpK9nhncQ6rzGxs9NSg3ygIpK2l4dc7tZpGOuxfyRpD" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{3x}{2}=π+2πn:{\\quad}x=\\frac{2π}{3}+\\frac{4πn}{3}$$", "input": "\\frac{3x}{2}=π+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{3x}{2}=π+2πn", "result": "3x=2π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:3x}{2}=2π+2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "3x=2π+4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3x=2π+4πn", "result": "x=\\frac{2π}{3}+\\frac{4πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3x}{3}=\\frac{2π}{3}+\\frac{4πn}{3}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{2π}{3}+\\frac{4πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"$$\\sin\\left(\\frac{5x}{2}\\right)=0{\\quad:\\quad}x=\\frac{4πn}{5},\\:x=\\frac{2π}{5}+\\frac{4πn}{5}$$", "input": "\\sin\\left(\\frac{5x}{2}\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(\\frac{5x}{2}\\right)=0$$", "result": "\\frac{5x}{2}=0+2πn,\\:\\frac{5x}{2}=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "\\frac{5x}{2}=0+2πn,\\:\\frac{5x}{2}=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{5x}{2}=0+2πn:{\\quad}x=\\frac{4πn}{5}$$", "input": "\\frac{5x}{2}=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "\\frac{5x}{2}=2πn" }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{5x}{2}=2πn", "result": "5x=4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:5x}{2}=2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "5x=4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": 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1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{5x}{2}=π+2πn:{\\quad}x=\\frac{2π}{5}+\\frac{4πn}{5}$$", "input": "\\frac{5x}{2}=π+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{5x}{2}=π+2πn", "result": "5x=2π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\cdot\\:5x}{2}=2π+2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "5x=2π+4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5x=2π+4πn", "result": "x=\\frac{2π}{5}+\\frac{4πn}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5x}{5}=\\frac{2π}{5}+\\frac{4πn}{5}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{2π}{5}+\\frac{4πn}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{4πn}{5},\\:x=\\frac{2π}{5}+\\frac{4πn}{5}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{4πn}{3},\\:x=\\frac{2π}{3}+\\frac{4πn}{3},\\:x=\\frac{4πn}{5},\\:x=\\frac{2π}{5}+\\frac{4πn}{5}" } ], "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)-\\cos(4x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }