2cos^2(x)=3cos(x)+2

{ "query": { "display": "$$2\\cos^{2}\\left(x\\right)=3\\cos\\left(x\\right)+2$$", "symbolab_question": "EQUATION#2\\cos^{2}(x)=3\\cos(x)+2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{2π}{3}+2πn,x=\\frac{4π}{3}+2πn", "degrees": "x=120^{\\circ }+360^{\\circ }n,x=240^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos^{2}\\left(x\\right)=3\\cos\\left(x\\right)+2{\\quad:\\quad}x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn$$", "input": "2\\cos^{2}\\left(x\\right)=3\\cos\\left(x\\right)+2", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "2\\cos^{2}\\left(x\\right)=3\\cos\\left(x\\right)+2", "result": "\\cos\\left(x\\right)=2,\\:\\cos\\left(x\\right)=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "2u^{2}=3u+2" }, { "type": "interim", "title": "$$2u^{2}=3u+2{\\quad:\\quad}u=2,\\:u=-\\frac{1}{2}$$", "input": "2u^{2}=3u+2", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the left side", "input": "2u^{2}=3u+2", "result": "2u^{2}-2=3u", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "2u^{2}-2=3u+2-2" }, { "type": "step", "primary": "Simplify", "result": "2u^{2}-2=3u" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$3u\\:$$to the left side", "input": "2u^{2}-2=3u", "result": "2u^{2}-2-3u=0", "steps": [ { "type": "step", "primary": "Subtract $$3u$$ from both sides", "result": "2u^{2}-2-3u=3u-3u" }, { "type": "step", "primary": "Simplify", "result": "2u^{2}-2-3u=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "2u^{2}-3u-2=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "2u^{2}-3u-2=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:2\\left(-2\\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 $${\\quad}a=2,\\:b=-3,\\:c=-2$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:2\\left(-2\\right)}}{2\\cdot\\:2}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:2\\left(-2\\right)}=5$$", "input": "\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:2\\left(-2\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:5}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-3\\right)^{2}+4\\cdot\\:2\\cdot\\:2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-3\\right)^{2}=3^{2}$$" ], "result": "=\\sqrt{3^{2}+4\\cdot\\:2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2\\cdot\\:2=16$$", "result": "=\\sqrt{3^{2}+16}" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\sqrt{9+16}" }, { "type": "step", "primary": "Add the numbers: $$9+16=25$$", "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77dtxLd9UeyppCvogsL4v8xo8YJJMq21DhvmKFx9fgtkAlilG71elit3w1IBbYN0PPMIzKJkow6rtuXltoCsjrqN6Hv6MoTMtvtU0IQwXdn8J9HDrbqRwJyIBoCY7JVoJrWtsflfjzKgQu80yQwgzBCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-3\\right)+5}{2\\cdot\\:2},\\:{u}_{2}=\\frac{-\\left(-3\\right)-5}{2\\cdot\\:2}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-3\\right)+5}{2\\cdot\\:2}:{\\quad}2$$", "input": "\\frac{-\\left(-3\\right)+5}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{3+5}{2\\cdot\\:2}" }, { "type": "step", "primary": "Add the numbers: $$3+5=8$$", "result": "=\\frac{8}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{8}{4}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{4}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sieDD8tdoqGWFRBUAjqnxpKet2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiKRcASOqRpLIeIyUBzgC+nUCr6EdYdh/n2c4DPMiuUGOLnd1TAgOBWmZVEULOgvgERx0ij68Lvr+79BVqXMQAj8w==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-3\\right)-5}{2\\cdot\\:2}:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{-\\left(-3\\right)-5}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{3-5}{2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$3-5=-2$$", "result": "=\\frac{-2}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{-2}{4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7q2cS8hAG+/7rXt5n0OkHv5Ket2LMwfmuCAWeFteXWHQgJ/ZZA32ZInFBpDtxBfiKmWiTEpQjat3SO7/l2m58l6mNsLv4gXj0VPO6vgHds8nFyoUUXuvn7kSsC/X/60sU0X9/eS2Lc0uU8wmW0XB2so8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=2,\\:u=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=2,\\:\\cos\\left(x\\right)=-\\frac{1}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=2{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)=2", "steps": [ { "type": "step", "primary": "$$-1\\le\\cos\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-\\frac{1}{2}{\\quad:\\quad}x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn$$", "input": "\\cos\\left(x\\right)=-\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=-\\frac{1}{2}$$", "result": "x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+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=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\cos^{2}(x)-3\\cos(x)-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }