cos(2x)-cos(x)=2

{ "query": { "display": "$$\\cos\\left(2x\\right)-\\cos\\left(x\\right)=2$$", "symbolab_question": "EQUATION#\\cos(2x)-\\cos(x)=2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=π+2πn", "degrees": "x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(2x\\right)-\\cos\\left(x\\right)=2{\\quad:\\quad}x=π+2πn$$", "input": "\\cos\\left(2x\\right)-\\cos\\left(x\\right)=2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "\\cos\\left(2x\\right)-\\cos\\left(x\\right)-2=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-2+\\cos\\left(2x\\right)-\\cos\\left(x\\right)", "result": "-3-\\cos\\left(x\\right)+2\\cos^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2x\\right)=2\\cos^{2}\\left(x\\right)-1$$", "result": "=-2+2\\cos^{2}\\left(x\\right)-1-\\cos\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$-2+2\\cos^{2}\\left(x\\right)-1-\\cos\\left(x\\right):{\\quad}2\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-3$$", "input": "-2+2\\cos^{2}\\left(x\\right)-1-\\cos\\left(x\\right)", "result": "=2\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-3", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=2\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-2-1" }, { "type": "step", "primary": "Subtract the numbers: $$-2-1=-3$$", "result": "=2\\cos^{2}\\left(x\\right)-\\cos\\left(x\\right)-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BYYZopJo7CEHl46zGb0PqjF88oZh36CRsu4+SNenPgTehkKrn0era9rz8TlL+x/vK681lBK6gAQJrscFq11Ga+7PFFRfVMsU6iNV3gu8rFtqOVWDvNoiJVNLOI88lUmTHimBRYRqHSWeJkuUPhfTC6X8DkMMm4VZoz6FnrJ/KhECSOQVazemDVebr6k6c0M4sIjaxJ4DvjTb2fbKjbvtlQ==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73hhjkMWe9cYcrFuqhYZiHw9fdMTlaJAeK2FKyc6j9Gw52FciCV6Q/ZuTzBHIPdDy6bFjjx5IuxvIy7/nJMTFRsLgcgrrcmkH6sZs7VXsfRBbEoBm75s615kjFQXZBnT3ciCGIMxXgHWE8NAQOjjBM5eWpcVui67RFsCVNGsGk5a9dDZkXCs5+Cxg1YNvJNPUe9yYF2Q5qSv3gDlLQoBFjj5yGUAwgawpYTH/Vvtg3vE=" } }, { "type": "interim", "title": "Solve by substitution", "input": "-3-\\cos\\left(x\\right)+2\\cos^{2}\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=\\frac{3}{2},\\:\\cos\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "-3-u+2u^{2}=0" }, { "type": "interim", "title": "$$-3-u+2u^{2}=0{\\quad:\\quad}u=\\frac{3}{2},\\:u=-1$$", "input": "-3-u+2u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "2u^{2}-u-3=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "2u^{2}-u-3=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:2\\left(-3\\right)}}{2\\cdot\\:2}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=2,\\:b=-1,\\:c=-3$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:2\\left(-3\\right)}}{2\\cdot\\:2}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:2\\left(-3\\right)}=5$$", "input": "\\sqrt{\\left(-1\\right)^{2}-4\\cdot\\:2\\left(-3\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-1\\right)\\pm\\:5}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-1\\right)^{2}+4\\cdot\\:2\\cdot\\:3}" }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "interim", "title": "$$4\\cdot\\:2\\cdot\\:3=24$$", "input": "4\\cdot\\:2\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2\\cdot\\:3=24$$", "result": "=24" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BrBjmPYwjyi388939n5bKRUbjr1up91GPxQqAij33CujkVi15I8rBefLi4Iyt2wrA5mNI8d524k0MHHn76yyMQ0IERTknIByea4Ue9OUx2oS+5NYM0KcSlqgXx1KKMc7" } }, { "type": "step", "result": "=\\sqrt{1+24}" }, { "type": "step", "primary": "Add the numbers: $$1+24=25$$", "result": "=\\sqrt{25}" }, { "type": "step", "primary": "Factor the number: $$25=5^{2}$$", "result": "=\\sqrt{5^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{5^{2}}=5$$" ], "result": "=5", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gcCSkwHGkSzBJS28+axO6+cXGfCIvHMp7DG0rtYxvSkAlilG71elit3w1IBbYN0PPMIzKJkow6rtuXltoCsjrqN6Hv6MoTMtvtU0IQwXdn8tVZVn3juMEEyoNr8u6sPO/20NCjjgtUFenvCeAenMZiS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-1\\right)+5}{2\\cdot\\:2},\\:{u}_{2}=\\frac{-\\left(-1\\right)-5}{2\\cdot\\:2}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-1\\right)+5}{2\\cdot\\:2}:{\\quad}\\frac{3}{2}$$", "input": "\\frac{-\\left(-1\\right)+5}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1+5}{2\\cdot\\:2}" }, { "type": "step", "primary": "Add the numbers: $$1+5=6$$", "result": "=\\frac{6}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{6}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{3}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rYEb5H2UhMed/LULBMRUu5Ket2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJxMNflh0wYFmC9/v0d6c7p6iuu1BULe4p4TCP1Dge8oRwY8ZsFOhASenMdRTOVctwYmpXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-1\\right)-5}{2\\cdot\\:2}:{\\quad}-1$$", "input": "\\frac{-\\left(-1\\right)-5}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1-5}{2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$1-5=-4$$", "result": "=\\frac{-4}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{-4}{4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xrubhzl7D6ENB059zrTkiJKet2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiKSECk6GWu1Zs/UVqyB86KAkfQ7pOlkQMojVPaIwryp7kuWwABEef3h6ykyFtgs/8JfHoXIQE40FG2d5xHqpEqgg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{3}{2},\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=\\frac{3}{2},\\:\\cos\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=\\frac{3}{2}{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=\\frac{3}{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\cos\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-1{\\quad:\\quad}x=π+2πn$$", "input": "\\cos\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=-1$$", "result": "x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(2x)-\\cos(x)-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }