3sec(x)+5-cos(x)=cos(x)

{ "query": { "display": "$$3\\sec\\left(x\\right)+5-\\cos\\left(x\\right)=\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#3\\sec(x)+5-\\cos(x)=\\cos(x)" }, "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": "$$3\\sec\\left(x\\right)+5-\\cos\\left(x\\right)=\\cos\\left(x\\right){\\quad:\\quad}x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn$$", "input": "3\\sec\\left(x\\right)+5-\\cos\\left(x\\right)=\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cos\\left(x\\right)$$ from both sides", "result": "3\\sec\\left(x\\right)+5-2\\cos\\left(x\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "5-2\\cos\\left(x\\right)+3\\sec\\left(x\\right)", "result": "5-\\frac{2}{\\sec\\left(x\\right)}+3\\sec\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\cos\\left(x\\right)=\\frac{1}{\\sec\\left(x\\right)}$$", "result": "=5-2\\cdot\\:\\frac{1}{\\sec\\left(x\\right)}+3\\sec\\left(x\\right)" }, { "type": "interim", "title": "$$2\\cdot\\:\\frac{1}{\\sec\\left(x\\right)}=\\frac{2}{\\sec\\left(x\\right)}$$", "input": "2\\cdot\\:\\frac{1}{\\sec\\left(x\\right)}", "result": "=5-\\frac{2}{\\sec\\left(x\\right)}+3\\sec\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{\\sec\\left(x\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:2=2$$", "result": "=\\frac{2}{\\sec\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviqgTrE5NivJpRVqDuGxbtozehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgpgfGsjZiCarqaPWfalgKfNl04xTNlEc8iWnVHxPamSnbS8PgbRY6lX6tcMqfLeRhC8+DQy55dl6Cq10f09Im2VESkcaBOQK+LJ5cxZWqY7oa" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OQ7Kq8bG1VdHDCUGNY+lE4cgCQHWCyeiw08tQwTtVFI52FciCV6Q/ZuTzBHIPdDy6bFjjx5IuxvIy7/nJMTFRsLgcgrrcmkH6sZs7VXsfRAaTV0Ia9XHcwoglh2xkm12lMNiIJPXgXTj9mGDfFMBNblEL1mi072Ww3mBUSWzk9xFKk3fejFkyiOiq9iG9IkAO/B7UtNBfw19bDRpX2AlzDEZOhzOLFFCWOVe0TAs9Ps/tVtg7k21KzJIFYRgfW7H" } }, { "type": "interim", "title": "Solve by substitution", "input": "5-\\frac{2}{\\sec\\left(x\\right)}+3\\sec\\left(x\\right)=0", "result": "\\sec\\left(x\\right)=\\frac{1}{3},\\:\\sec\\left(x\\right)=-2", "steps": [ { "type": "step", "primary": "Let: $$\\sec\\left(x\\right)=u$$", "result": "5-\\frac{2}{u}+3u=0" }, { "type": "interim", "title": "$$5-\\frac{2}{u}+3u=0{\\quad:\\quad}u=\\frac{1}{3},\\:u=-2$$", "input": "5-\\frac{2}{u}+3u=0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$u$$", "input": "5-\\frac{2}{u}+3u=0", "result": "5u-2+3u^{2}=0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$u$$", "result": "5u-\\frac{2}{u}u+3uu=0\\cdot\\:u" }, { "type": "interim", "title": "Simplify", "input": "5u-\\frac{2}{u}u+3uu=0\\cdot\\:u", "result": "5u-2+3u^{2}=0", "steps": [ { "type": "interim", "title": "Simplify $$-\\frac{2}{u}u:{\\quad}-2$$", "input": "-\\frac{2}{u}u", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{2u}{u}" }, { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UYGzpPoE9vxuUNTd5y+OTgCWKUbvV6WK3fDUgFtg3Q+Ts+e1RE2erKWpqsVv3YjHZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0xQsyPybNKiF9s6AfieQLQsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$3uu:{\\quad}3u^{2}$$", "input": "3uu", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=3u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=3u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7dwTCeLSVwiafPzRfrO0CsXCQoYlYQ8U+Tfyx0kyzI8hsXOqmTUp5XGSMGN7hpWsreqXxdc+rps1CUyb7fqI2GWh46iFhxZGPboI5cR4UGpwkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Simplify $$0\\cdot\\:u:{\\quad}0$$", "input": "0\\cdot\\:u", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Z1CIVc4HWO3x8GamLSkHTt13jtrSFDx+UNsawjlOjV1NsA7FPzA5OWzDmAZ/4d2v1sD7NfhsPe7eDHrmjY0mE7fcgUYIoNC9L37RdnKXleh/c2B53pG/u90QsOfZeCsO" } }, { "type": "step", "result": "5u-2+3u^{2}=0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve $$5u-2+3u^{2}=0:{\\quad}u=\\frac{1}{3},\\:u=-2$$", "input": "5u-2+3u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "3u^{2}+5u-2=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "3u^{2}+5u-2=0", "result": "{u}_{1,\\:2}=\\frac{-5\\pm\\:\\sqrt{5^{2}-4\\cdot\\:3\\left(-2\\right)}}{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 $${\\quad}a=3,\\:b=5,\\:c=-2$$", "result": "{u}_{1,\\:2}=\\frac{-5\\pm\\:\\sqrt{5^{2}-4\\cdot\\:3\\left(-2\\right)}}{2\\cdot\\:3}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{5^{2}-4\\cdot\\:3\\left(-2\\right)}=7$$", "input": "\\sqrt{5^{2}-4\\cdot\\:3\\left(-2\\right)}", "result": "{u}_{1,\\:2}=\\frac{-5\\pm\\:7}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{5^{2}+4\\cdot\\:3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3\\cdot\\:2=24$$", "result": "=\\sqrt{5^{2}+24}" }, { "type": "step", "primary": "$$5^{2}=25$$", "result": "=\\sqrt{25+24}" }, { "type": "step", "primary": "Add the numbers: $$25+24=49$$", "result": "=\\sqrt{49}" }, { "type": "step", "primary": "Factor the number: $$49=7^{2}$$", "result": "=\\sqrt{7^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{7^{2}}=7$$" ], "result": "=7", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74f6cUYKncTCZw7Lf+vpnnthv2mZYI8kY7XDGEtWJLWzehkKrn0era9rz8TlL+x/vTE4OONfGJ7J2yqTcD4S3brtCR5dIjxQ5ASg+ZPFVSsddr7Pkd5z1di9LpSFWP/H3CJtUnPKx9LKD5cZoY4LCFA==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-5+7}{2\\cdot\\:3},\\:{u}_{2}=\\frac{-5-7}{2\\cdot\\:3}" }, { "type": "interim", "title": "$$u=\\frac{-5+7}{2\\cdot\\:3}:{\\quad}\\frac{1}{3}$$", "input": "\\frac{-5+7}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-5+7=2$$", "result": "=\\frac{2}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{2}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79my7jr4R5Z/4c9+xxlxXS8EEAaPuLxfWvDabf+CrYVEDnzlbPZjyKgy1eUCFsLd5xXDODtFBCC8Uf836IcE9xxSYeD6JDEaUbEpYNvzKYnPI339UdswAuo1+g3Q7EWAYtoii6Ip2Ifcb1STKHSt/nQ==" } }, { "type": "interim", "title": "$$u=\\frac{-5-7}{2\\cdot\\:3}:{\\quad}-2$$", "input": "\\frac{-5-7}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-5-7=-12$$", "result": "=\\frac{-12}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{-12}{6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{12}{6}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{12}{6}=2$$", "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ofMrj6GWSJ0om++n4Fw9FsEEAaPuLxfWvDabf+CrYVEDnzlbPZjyKgy1eUCFsLd5RDtWG5guT8Xmwx3nth5Sgj/AeYEIX4I11Ba3Yz03hSAzWvmoXNt/mwyREh1/I3A/ialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{1}{3},\\:u=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=\\frac{1}{3},\\:u=-2" }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}u=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$5-\\frac{2}{u}+3u$$ and compare to zero" }, { "type": "step", "result": "u=0" }, { "type": "step", "primary": "The following points are undefined", "result": "u=0" } ], "meta": { "interimType": "Undefined Points 0Eq" } }, { "type": "step", "primary": "Combine undefined points with solutions:" }, { "type": "step", "result": "u=\\frac{1}{3},\\:u=-2" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(x\\right)$$", "result": "\\sec\\left(x\\right)=\\frac{1}{3},\\:\\sec\\left(x\\right)=-2" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sec\\left(x\\right)=\\frac{1}{3}{\\quad:\\quad}$$No Solution", "input": "\\sec\\left(x\\right)=\\frac{1}{3}", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)\\le-1\\lor\\sec\\left(x\\right)\\ge1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sec\\left(x\\right)=-2{\\quad:\\quad}x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn$$", "input": "\\sec\\left(x\\right)=-2", "steps": [ { "type": "interim", "title": "General solutions for $$\\sec\\left(x\\right)=-2$$", "result": "x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{2π}{3}+2πn,\\:x=\\frac{4π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions sec 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": "3\\sec(x)+5-\\cos(x)-\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }