cos(x)-cos(3x)=-sin(x)

{ "query": { "display": "$$\\cos\\left(x\\right)-\\cos\\left(3x\\right)=-\\sin\\left(x\\right)$$", "symbolab_question": "EQUATION#\\cos(x)-\\cos(3x)=-\\sin(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn,x=\\frac{7π}{12}+πn,x=\\frac{11π}{12}+πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n,x=105^{\\circ }+180^{\\circ }n,x=165^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(x\\right)-\\cos\\left(3x\\right)=-\\sin\\left(x\\right){\\quad:\\quad}x=2πn,\\:x=π+2πn,\\:x=\\frac{7π}{12}+πn,\\:x=\\frac{11π}{12}+πn$$", "input": "\\cos\\left(x\\right)-\\cos\\left(3x\\right)=-\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$-\\sin\\left(x\\right)$$ from both sides", "result": "\\cos\\left(x\\right)-\\cos\\left(3x\\right)+\\sin\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-\\cos\\left(3x\\right)+\\cos\\left(x\\right)+\\sin\\left(x\\right)", "result": "\\sin\\left(x\\right)+2\\sin\\left(2x\\right)\\sin\\left(x\\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": "=\\sin\\left(x\\right)-2\\sin\\left(\\frac{x+3x}{2}\\right)\\sin\\left(\\frac{x-3x}{2}\\right)" }, { "type": "interim", "title": "Simplify $$\\sin\\left(x\\right)-2\\sin\\left(\\frac{x+3x}{2}\\right)\\sin\\left(\\frac{x-3x}{2}\\right):{\\quad}\\sin\\left(x\\right)+2\\sin\\left(x\\right)\\sin\\left(2x\\right)$$", "input": "\\sin\\left(x\\right)-2\\sin\\left(\\frac{x+3x}{2}\\right)\\sin\\left(\\frac{x-3x}{2}\\right)", "result": "=\\sin\\left(x\\right)+2\\sin\\left(x\\right)\\sin\\left(2x\\right)", "steps": [ { "type": "interim", "title": "$$2\\sin\\left(\\frac{x+3x}{2}\\right)\\sin\\left(\\frac{x-3x}{2}\\right)=-2\\sin\\left(x\\right)\\sin\\left(2x\\right)$$", "input": "2\\sin\\left(\\frac{x+3x}{2}\\right)\\sin\\left(\\frac{x-3x}{2}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{x+3x}{2}=2x$$", "input": "\\frac{x+3x}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$x+3x=4x$$", "result": "=\\frac{4x}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7idDRn/iFPHhF94efrbwHzXyRHuGw7+tM5METTDj6vVFNBj1IDrq1hA5sUPhrfJHOP8vQyhiD4JSfqjIvcQ7tivoHwqV2NpguKrQEBKNvH+hHyk8Sw2LNMPalzST86pQ2" } }, { "type": "step", "result": "=2\\sin\\left(2x\\right)\\sin\\left(\\frac{x-3x}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{x-3x}{2}=-x$$", "input": "\\frac{x-3x}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$x-3x=-2x$$", "result": "=\\frac{-2x}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2x}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=-x" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7x4C4UfngGZDiLvN7alXzlXyRHuGw7+tM5METTDj6vVEGuSvrBTGv/RcBUE2duDFVP8vQyhiD4JSfqjIvcQ7tilrav05H+sfTGRH5DVwjgNk8E/GTbvUmQTH86aCFEwWM" } }, { "type": "step", "result": "=2\\sin\\left(2x\\right)\\sin\\left(-x\\right)" }, { "type": "step", "primary": "Use the negative angle identity: $$\\sin\\left(-x\\right)=-\\sin\\left(x\\right)$$", "secondary": [], "result": "=2\\left(-\\sin\\left(x\\right)\\right)\\sin\\left(2x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EShBlh31jWu86BCY6M6r99VBRVSHxZIvzR5jT91JeEu/7ypHTUrv495Ot28u+bqNLTrWWMFI8l4Q07DZ5+hJa7Q5dMaHni/ZoCTXF8I763un80Il31u0IfyMu7ezCGBv7lvKtANBUJdQPS8f9+853MDw+ZLM05d0+f3Ba72V64g4zXA2qwg9xFFFo/9ObELqSi8NhG5v+lRJKb4v/kmGOm7O+3EQLht9YQtiDiv4UnfzEbe6tY97b8HX6Oo5G/x2" } }, { "type": "step", "result": "=\\sin\\left(x\\right)-\\left(-2\\sin\\left(x\\right)\\sin\\left(2x\\right)\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sin\\left(x\\right)+2\\sin\\left(x\\right)\\sin\\left(2x\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xOiZQOhs5FXW6FC50wgs0F8qbza7cTByzA4QeQhCbisKfDq+/XhdA63850xgvCTfmlo/75Ia8gen0rh5j1LLVHWD310L1+P2yDQQfMEhENGt3u8g9Q2FtxtjFDzn32MsDGAMPNU6AWnPHJIu7zTz9T/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10Jkqelwu+mS6hPShOorHPG0+JAmDBJb8G0SgE41QHBy/xyJfNgCgIjt8CNHxYiwSunOwi7eiuYZ5PUbsF8UEeAg==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sxSOfOl7Um+zqSbarNITtufTnyj2QJPbTwd8xIM42MRFKEhauahU1WePKtfAwwZ/glS0lOYo+MEX6Bpkgk4kWW2OqI7u8uw/2TDZOKR8MlHIjRu93pF6Ud8rAuy8zIsmAr63JSvw+N5wTI9TqH9lx8FERmT1Ud8ID0bE6ityN/0hpPLyCYrLk9jd6X5FtSaWSYU/rlY9QcIgZOnNP8QBxx62DrT5lSWo/mMH5F6V1QDkAxHFda36329rSpnJqDHJ" } }, { "type": "interim", "title": "Factor $$\\sin\\left(x\\right)+2\\sin\\left(2x\\right)\\sin\\left(x\\right):{\\quad}\\sin\\left(x\\right)\\left(2\\sin\\left(2x\\right)+1\\right)$$", "input": "\\sin\\left(x\\right)+2\\sin\\left(2x\\right)\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$\\sin\\left(x\\right)$$", "result": "=\\sin\\left(x\\right)\\left(1+2\\sin\\left(2x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\sin\\left(x\\right)\\left(2\\sin\\left(2x\\right)+1\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(x\\right)=0\\lor\\:2\\sin\\left(2x\\right)+1=0" }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=0$$", "result": "x=0+2πn,\\:x=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": "x=0+2πn,\\:x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$2\\sin\\left(2x\\right)+1=0{\\quad:\\quad}x=\\frac{7π}{12}+πn,\\:x=\\frac{11π}{12}+πn$$", "input": "2\\sin\\left(2x\\right)+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "2\\sin\\left(2x\\right)+1=0", "result": "2\\sin\\left(2x\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "2\\sin\\left(2x\\right)+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "2\\sin\\left(2x\\right)=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\sin\\left(2x\\right)=-1", "result": "\\sin\\left(2x\\right)=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\sin\\left(2x\\right)}{2}=\\frac{-1}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(2x\\right)=-\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ekOAcPFfzmZkcNOvaCOLEQJ2gLsqlksUAODBZimNRkx32tovqQJUqmDj6vwvJQPSgd8+3Xo2a60a2d55q+FOoiH8y7DwPH3omw6BVDpsKavYLUU9f55113qmoXc/qYUoewjtgv2P/LGLkY0QNp+zPs9u8aYonAi5Xo/q9yqqYOqzpwyw9I7LHVQgxmET7aFUQQKn7/Khp+NH0cZzyTOxqCjvnfPVdJLFLCOMX+1kqONWtB9wKnsf5AhfjFfBk4LYdZSKc6rDuAki8hWkk9yv6KfIvMqT//VOD98zkDccMkK6Zn2Hj/sMR076HKJpxwGubM/a2jlmVmIdR4juY5lyqfEqyhRW2Vgp0dJqPBUClzh5HMze/KerEkoiDeO0X1kJj51HCRqdlolTDJhJwLVMLO+VSz2yw3ga51Ab5U4/cafK3wsVCblSmxuxnmJ9OSEUZGkgmoxoNlYnAOkblJbByx3tkbpEhAhHqWbUSSXvVLtFKk3fejFkyiOiq9iG9IkAybeXGnJZsE3KymoroUC4qqd6FWO7orOoazvWOuESp7U=" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(2x\\right)=-\\frac{1}{2}$$", "result": "2x=\\frac{7π}{6}+2πn,\\:2x=\\frac{11π}{6}+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=\\frac{7π}{6}+2πn,\\:2x=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{7π}{6}+2πn:{\\quad}x=\\frac{7π}{12}+πn$$", "input": "2x=\\frac{7π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{7π}{6}+2πn", "result": "x=\\frac{7π}{12}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{7π}{12}+πn", "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 $$\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{7π}{12}+πn$$", "input": "\\frac{\\frac{7π}{6}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{7π}{6}}{2}=\\frac{7π}{12}$$", "input": "\\frac{\\frac{7π}{6}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{7π}{6\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\frac{7π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXlEtnBRHsSpEpm9woUlEmbdd47a0hQ8flDbGsI5To1dVESQA48qouUS12usnKDA2ELhcJuE223isn9fcbq91bYjigtccxNWJk7pEnO8OW1UTwuOeq45ZybiEP9otuZjpKuNnDpNe1DOk1tOxIo/dxawiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{7π}{12}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajeqNE4F/56WSLxMRCLVPLIi2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXVREkAOPKqLlEtdrrJygwNjmV2RD9WIotqtX60WMCq0CgQUxJPyUNnGfVirkcwpVO0BvnUFx4T5rmmmJUxss5NnvAdonft/6Id5YgksLuZ342RlP6nbKbEC/HotWxNZG8pW9+QW0/o5wsTggtcymdGQ=" } }, { "type": "step", "result": "x=\\frac{7π}{12}+πn" } ], "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 $$2x=\\frac{11π}{6}+2πn:{\\quad}x=\\frac{11π}{12}+πn$$", "input": "2x=\\frac{11π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{11π}{6}+2πn", "result": "x=\\frac{11π}{12}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}", "result": "x=\\frac{11π}{12}+πn", "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 $$\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{11π}{12}+πn$$", "input": "\\frac{\\frac{11π}{6}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{11π}{6}}{2}=\\frac{11π}{12}$$", "input": "\\frac{\\frac{11π}{6}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{11π}{6\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:2=12$$", "result": "=\\frac{11π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajcm5FXQ50IQsy9V6MeVoA3UgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJ4AbY1ZZlr0ES7ShAvPM/yhTW26qciuyUBGXQExCUedYvS8N5VCze8qz+LNUf5QAR8NqfIbgCoEQ5iqOpxvXCtHuJ1iXawr0xn/4zW2wj+oZ" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{11π}{12}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88aja8YN6VzLM/g2fVQvpHYB3AYKXMStDfJcV1gA+Z/JDXIICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyfmx8MSjQWWYpQEOqS7iNsYRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6q99iKcg75ivhzP3grxubPxeAToLcJKuKIp3DLH8FrZfjkRYT57po0fVp+kBBk2foE=" } }, { "type": "step", "result": "x=\\frac{11π}{12}+πn" } ], "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=\\frac{7π}{12}+πn,\\:x=\\frac{11π}{12}+πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{7π}{12}+πn,\\:x=\\frac{11π}{12}+πn" } ], "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(3x)+\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }