cos(4x)=cos(8x)

{ "query": { "display": "$$\\cos\\left(4x\\right)=\\cos\\left(8x\\right)$$", "symbolab_question": "EQUATION#\\cos(4x)=\\cos(8x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=πn,x=\\frac{π}{2}+πn,x=\\frac{πn}{3},x=\\frac{π}{6}+\\frac{πn}{3}", "degrees": "x=0^{\\circ }+180^{\\circ }n,x=90^{\\circ }+180^{\\circ }n,x=0^{\\circ }+60^{\\circ }n,x=30^{\\circ }+60^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(4x\\right)=\\cos\\left(8x\\right){\\quad:\\quad}x=πn,\\:x=\\frac{π}{2}+πn,\\:x=\\frac{πn}{3},\\:x=\\frac{π}{6}+\\frac{πn}{3}$$", "input": "\\cos\\left(4x\\right)=\\cos\\left(8x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cos\\left(8x\\right)$$ from both sides", "result": "\\cos\\left(4x\\right)-\\cos\\left(8x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(4x\\right)-\\cos\\left(8x\\right)", "result": "2\\sin\\left(2x\\right)\\sin\\left(6x\\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{4x+8x}{2}\\right)\\sin\\left(\\frac{4x-8x}{2}\\right)" }, { "type": "interim", "title": "Simplify $$-2\\sin\\left(\\frac{4x+8x}{2}\\right)\\sin\\left(\\frac{4x-8x}{2}\\right):{\\quad}2\\sin\\left(2x\\right)\\sin\\left(6x\\right)$$", "input": "-2\\sin\\left(\\frac{4x+8x}{2}\\right)\\sin\\left(\\frac{4x-8x}{2}\\right)", "result": "=2\\sin\\left(2x\\right)\\sin\\left(6x\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{4x+8x}{2}=6x$$", "input": "\\frac{4x+8x}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$4x+8x=12x$$", "result": "=\\frac{12x}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{2}=6$$", "result": "=6x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bI6Dn7lC3ZrNGbfD8VQ7+lnyYRz18HvB+rp63mPitc/smK6PJrHKQtHtO/Q2v7A4NnlcfYSxPIgsp1FH5TTQYCkaP9RLrGVZS+Xv/sB9CUqDbbimLoQoheRZNhOz9WU3" } }, { "type": "step", "result": "=-2\\sin\\left(6x\\right)\\sin\\left(\\frac{4x-8x}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{4x-8x}{2}=-2x$$", "input": "\\frac{4x-8x}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$4x-8x=-4x$$", "result": "=\\frac{-4x}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4x}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=-2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MA/amQqDM0heVjwSCtiTX1nyYRz18HvB+rp63mPitc95tMpJTBBccUWkSyvMe1Sp2GIoxg2Jpr0LaZPQ02JWI9TbNeUUeWIvdEt5+FfB0f96ibeo9GOzpDpHjEyMEBqL" } }, { "type": "step", "result": "=-2\\sin\\left(6x\\right)\\sin\\left(-2x\\right)" }, { "type": "step", "primary": "Use the negative angle identity: $$\\sin\\left(-x\\right)=-\\sin\\left(x\\right)$$", "secondary": [], "result": "=-2\\left(-\\sin\\left(2x\\right)\\right)\\sin\\left(6x\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=2\\sin\\left(2x\\right)\\sin\\left(6x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WKQDtlNbtfOFijcUoWrkOgGxVKg1YmK9Lr22xHabJ8+A1BxDKxER34hqEQ20LZ7F5jGzK37+8I/eYs6AQMGtJ7OwcElXShGEqp/uHygEKBqZ/y3FyG+VW/X1FwY2AG4vcrjssvQjbL07oKLqRrsvHR4pgUWEah0lniZLlD4X0wvMvKiP8Qz916RQB8bZZchGV7CTrfnx04eHuAxZplVYk0jFXGNEtvzTWoNFS5sPz3FLiF4Mzbt6swySWFVTee4m" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbFfcMb18Lr2S4IENSjyIx0oFMS0G0HgjUXwc3RQOmn9BYznLdIBaHD4ibBnEILJTDXUHW1mE5UQUTIwuQd6ag+xNQUndUJlCvVvSnNUC8d6vTuAJsWGdGtnjPKQdLUcrb2RLd2VwIqlBNByF6663sySYtHHZIJ8Nhz40VaXBzMWWyHImFc42gKJRGjZFTjbQQUaaUzYDBaJ7PWzJE8yB6KA=" } }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(2x\\right)=0\\lor\\:\\sin\\left(6x\\right)=0" }, { "type": "interim", "title": "$$\\sin\\left(2x\\right)=0{\\quad:\\quad}x=πn,\\:x=\\frac{π}{2}+πn$$", "input": "\\sin\\left(2x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2x\\right)=0$$", "result": "2x=0+2πn,\\:2x=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": "2x=0+2πn,\\:2x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2x=0+2πn:{\\quad}x=πn$$", "input": "2x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "2x=2πn" }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=2πn", "result": "x=πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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"\\frac{2x}{2}=\\frac{π}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{2}+πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7s+VDSy9Nl5n0ZrAJ3ZUcLvdVtrzNWwcLBNloZMNuEPDya/gmXRIc30tREPIIyab8RoVJA0H1fnxI+eKaoiWe2vVBR0+98CcmmVtl3U0SxOdktDoFNMUm3jC8/4NQOpcsvFZh+W/iDMDAo4iCI7tFUdwpqaFhGel8eqIiHymEdPNIrutd8eSMgUo+HpRyFwAMBfv41OXHozr5g65Kr4Ll9cxQp8yk2EdviTvfhz6ruO+wCd2spQccAJgFCfgU1rRmuds+S4tbfrHLGkSh5QZa+g1AW5CvrMBr9GukingfRp3l9XY+6TAFvPNcX5SNrM6Ot7Z6xpkV0XHaGfuZNgIBdeCDCH2DcDNialMuvy7Bnwnk+UY9JBE5v/YSZ6s+GczznYqnACvNSqOPyqchT4gmfHabZwi5JytN/1ufmQiSimkGMgSS9o9O+UehR1a3T1ITWnYMM7mOiW9HKIEiHilarlHG4pyteBqJ+X90AXJ/ySqcAUskP3/0w3xzbPajQy3yKXGT2ZvqVP5WwVfXop0HoX4b2/ZopqwKGBXrlVbxxNY=" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=πn,\\:x=\\frac{π}{2}+πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sin\\left(6x\\right)=0{\\quad:\\quad}x=\\frac{πn}{3},\\:x=\\frac{π}{6}+\\frac{πn}{3}$$", "input": "\\sin\\left(6x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(6x\\right)=0$$", "result": "6x=0+2πn,\\:6x=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": "6x=0+2πn,\\:6x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$6x=0+2πn:{\\quad}x=\\frac{πn}{3}$$", "input": "6x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "6x=2πn" }, { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6x=2πn", "result": "x=\\frac{πn}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6x}{6}=\\frac{2πn}{6}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{πn}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": 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$$6$$", "result": "\\frac{6x}{6}=\\frac{π}{6}+\\frac{2πn}{6}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{π}{6}+\\frac{πn}{3}" } ], "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{πn}{3},\\:x=\\frac{π}{6}+\\frac{πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=πn,\\:x=\\frac{π}{2}+πn,\\:x=\\frac{πn}{3},\\:x=\\frac{π}{6}+\\frac{πn}{3}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(4x)-\\cos(8x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }