2-2cos^2(x)+3cos(x)=0

{ "query": { "display": "$$2-2\\cos^{2}\\left(x\\right)+3\\cos\\left(x\\right)=0$$", "symbolab_question": "EQUATION#2-2\\cos^{2}(x)+3\\cos(x)=0" }, "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": "$$2-2\\cos^{2}\\left(x\\right)+3\\cos\\left(x\\right)=0{\\quad:\\quad}x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn$$", "input": "2-2\\cos^{2}\\left(x\\right)+3\\cos\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "2-2\\cos^{2}\\left(x\\right)+3\\cos\\left(x\\right)=0", "result": "\\cos\\left(x\\right)=-\\frac{1}{2},\\:\\cos\\left(x\\right)=2", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "2-2u^{2}+3u=0" }, { "type": "interim", "title": "$$2-2u^{2}+3u=0{\\quad:\\quad}u=-\\frac{1}{2},\\:u=2$$", "input": "2-2u^{2}+3u=0", "steps": [ { "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{-3\\pm\\:\\sqrt{3^{2}-4\\left(-2\\right)\\cdot\\:2}}{2\\left(-2\\right)}", "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{-3\\pm\\:\\sqrt{3^{2}-4\\left(-2\\right)\\cdot\\:2}}{2\\left(-2\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{3^{2}-4\\left(-2\\right)\\cdot\\:2}=5$$", "input": "\\sqrt{3^{2}-4\\left(-2\\right)\\cdot\\:2}", "result": "{u}_{1,\\:2}=\\frac{-3\\pm\\:5}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I8nbFv5ohMEv/aq9F0opf5zWq2050FjMciA5vB2Acz3ehkKrn0era9rz8TlL+x/vQKNEobJS0nR/1Qk4oH0B/rtCR5dIjxQ5ASg+ZPFVSsejCsAfucb3rlKY34Y0DmcL1/WQx4hMbqbCs6E5fo55Jw==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-3+5}{2\\left(-2\\right)},\\:{u}_{2}=\\frac{-3-5}{2\\left(-2\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-3+5}{2\\left(-2\\right)}:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{-3+5}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-3+5}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Add/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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7gz2CS/b4lqGCI7mRS6Y821BraIXtDlgD3G/CwhQUjohwkKGJWEPFPk38sdJMsyPIg1TKcTLfFp/3mLqGU11tTy359V2bs1q67pXx94q5XArR1nvSom3iKA6XCKrL9V7tniE1DZR4gIVo9uGClUagGw==" } }, { "type": "interim", "title": "$$u=\\frac{-3-5}{2\\left(-2\\right)}:{\\quad}2$$", "input": "\\frac{-3-5}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-3-5}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract 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": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{8}{4}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{8}{4}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7O8JMJCHaeWfSWdzdmBLiN1BraIXtDlgD3G/CwhQUjohwkKGJWEPFPk38sdJMsyPIumUxCcUvFDU7OpAOYLk8aqYtRusONLQIGvnokQq3x8DxfayEPhINvNr8uCW/1LTC" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-\\frac{1}{2},\\:u=2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=-\\frac{1}{2},\\:\\cos\\left(x\\right)=2" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "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": "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": "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": "2-2\\cos^{2}(x)+3\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }