3cos^2(x)+4cos(x)+1=0

{ "query": { "display": "$$3\\cos^{2}\\left(x\\right)+4\\cos\\left(x\\right)+1=0$$", "symbolab_question": "EQUATION#3\\cos^{2}(x)+4\\cos(x)+1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=1.91063…+2πn,x=-1.91063…+2πn,x=π+2πn", "degrees": "x=109.47122…^{\\circ }+360^{\\circ }n,x=-109.47122…^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$3\\cos^{2}\\left(x\\right)+4\\cos\\left(x\\right)+1=0{\\quad:\\quad}x=1.91063…+2πn,\\:x=-1.91063…+2πn,\\:x=π+2πn$$", "input": "3\\cos^{2}\\left(x\\right)+4\\cos\\left(x\\right)+1=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "3\\cos^{2}\\left(x\\right)+4\\cos\\left(x\\right)+1=0", "result": "\\cos\\left(x\\right)=-\\frac{1}{3},\\:\\cos\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "3u^{2}+4u+1=0" }, { "type": "interim", "title": "$$3u^{2}+4u+1=0{\\quad:\\quad}u=-\\frac{1}{3},\\:u=-1$$", "input": "3u^{2}+4u+1=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "3u^{2}+4u+1=0", "result": "{u}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:3\\cdot\\:1}}{2\\cdot\\:3}", "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 $${\\quad}a=3,\\:b=4,\\:c=1$$", "result": "{u}_{1,\\:2}=\\frac{-4\\pm\\:\\sqrt{4^{2}-4\\cdot\\:3\\cdot\\:1}}{2\\cdot\\:3}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bPMyneU6LXHviR2M5I4mDOyk29n4cPq2sQpyMSmLlluo/JI5bBgpgExN510TA5cywo6iOxOQeL70yTMBl67ndVrsdW4QlnVHZaYhaD1DKw/yHO39OaqAj2uw1AILHkpQRCjpV70OAZjfZIQCCrJD3eiZ1i3SlIXIYyUwGbM2rffUZncC0SyrBAcpqUmpoHGwsBNFSR9QXznbEHhsIK4fR1PYpOViDRq7FqLS80u/wYzIpRtg1KjKz/V6OeceyrawCNPis6LyqkJxSUnzhCsmiw6nIt+eYKUUPD00vgtS92dHNZmI63uTFn0XtwBppBPxjkPVKIUPkodcPn/MaNclMcUTurTbVrqo2vHQhe+tTwiqQF9+99wH+RB4Hxq9JLbhHKpUotCYrgrrc84rzdFi/KLUsCNhqxeYt1i366MylsZ5HMze/KerEkoiDeO0X1kJj51HCRqdlolTDJhJwLVMLO+VSz2yw3ga51Ab5U4/cafK3wsVCblSmxuxnmJ9OSEUZGkgmoxoNlYnAOkblJbByx3tkbpEhAhHqWbUSSXvVLtFKk3fejFkyiOiq9iG9IkAkK4le0f/h2/I97ZbNwAJwgmBRhN5Vwyd4eFtz3EYGOKMAyxYyd6dEqfPuwJ4V8nZ" } }, { "type": "interim", "title": "$$\\sqrt{4^{2}-4\\cdot\\:3\\cdot\\:1}=2$$", "input": "\\sqrt{4^{2}-4\\cdot\\:3\\cdot\\:1}", "result": "{u}_{1,\\:2}=\\frac{-4\\pm\\:2}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3\\cdot\\:1=12$$", "result": "=\\sqrt{4^{2}-12}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=\\sqrt{16-12}" }, { "type": "step", "primary": "Subtract the numbers: $$16-12=4$$", "result": "=\\sqrt{4}" }, { "type": "step", "primary": "Factor the number: $$4=2^{2}$$", "result": "=\\sqrt{2^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74jz8pBgj569lwUyDg0Z0NtfWd2FGy381z/o+s7ovBJ18kR7hsO/rTOTBE0w4+r1R+icEvaaPC8duOntO4O50zGRLd2VwIqlBNByF6663syTlbd5UL5ZztzwKHGYuXpz+jRGI3cfN/9hEsVDVQLmpc7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-4+2}{2\\cdot\\:3},\\:{u}_{2}=\\frac{-4-2}{2\\cdot\\:3}" }, { "type": "interim", "title": "$$u=\\frac{-4+2}{2\\cdot\\:3}:{\\quad}-\\frac{1}{3}$$", "input": "\\frac{-4+2}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-4+2=-2$$", "result": "=\\frac{-2}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{-2}{6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73vGntpnK2uErnEdXIVpxusEEAaPuLxfWvDabf+CrYVEDnzlbPZjyKgy1eUCFsLd5+v0H1qOjS3ujgumCixKEvGYYnHLVfzWgXhHqK+R1GFr4w3jAXYYNLkhEssen6YAqrPD6QG5jA4SmAdY1VKKDpA==" } }, { "type": "interim", "title": "$$u=\\frac{-4-2}{2\\cdot\\:3}:{\\quad}-1$$", "input": "\\frac{-4-2}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-4-2=-6$$", "result": "=\\frac{-6}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{-6}{6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{6}{6}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zyCTmU6brsaMckwTiVVf1sEEAaPuLxfWvDabf+CrYVEDnzlbPZjyKgy1eUCFsLd5pmhpfHqzxU6UVoC7gLWdYz/AeYEIX4I11Ba3Yz03hSB/XRM30yHst8MR+NmehwQqialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-\\frac{1}{3},\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=-\\frac{1}{3},\\:\\cos\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-\\frac{1}{3}{\\quad:\\quad}x=\\arccos\\left(-\\frac{1}{3}\\right)+2πn,\\:x=-\\arccos\\left(-\\frac{1}{3}\\right)+2πn$$", "input": "\\cos\\left(x\\right)=-\\frac{1}{3}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(x\\right)=-\\frac{1}{3}", "result": "x=\\arccos\\left(-\\frac{1}{3}\\right)+2πn,\\:x=-\\arccos\\left(-\\frac{1}{3}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(x\\right)=-\\frac{1}{3}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "x=\\arccos\\left(-\\frac{1}{3}\\right)+2πn,\\:x=-\\arccos\\left(-\\frac{1}{3}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "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=\\arccos\\left(-\\frac{1}{3}\\right)+2πn,\\:x=-\\arccos\\left(-\\frac{1}{3}\\right)+2πn,\\:x=π+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=1.91063…+2πn,\\:x=-1.91063…+2πn,\\: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": "3\\cos^{2}(x)+4\\cos(x)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }