3sec(x)-cos(x)-2=0

{ "query": { "display": "$$3\\sec\\left(x\\right)-\\cos\\left(x\\right)-2=0$$", "symbolab_question": "EQUATION#3\\sec(x)-\\cos(x)-2=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$3\\sec\\left(x\\right)-\\cos\\left(x\\right)-2=0{\\quad:\\quad}x=2πn$$", "input": "3\\sec\\left(x\\right)-\\cos\\left(x\\right)-2=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "-2-\\cos\\left(x\\right)+3\\sec\\left(x\\right)", "result": "-2-\\frac{1}{\\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": "=-2-\\frac{1}{\\sec\\left(x\\right)}+3\\sec\\left(x\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+EuaasxbJOzJUokzfi4hmIcgCQHWCyeiw08tQwTtVFI52FciCV6Q/ZuTzBHIPdDy6bFjjx5IuxvIy7/nJMTFRsLgcgrrcmkH6sZs7VXsfRDyp041I7V1o5vGeUJRJvqZ3zELwzSUlxUgyZuZ1xpJUYt55OEv+hqf53zJIQZ8lWejeh7+jKEzLb7VNCEMF3Z/AUodDpEZVCOJY6rkfXcVkqaOQILKf053R9DIJaaWk7Mc6tftTNY2KsUtvVbis5zN" } }, { "type": "interim", "title": "Solve by substitution", "input": "-2-\\frac{1}{\\sec\\left(x\\right)}+3\\sec\\left(x\\right)=0", "result": "\\sec\\left(x\\right)=1,\\:\\sec\\left(x\\right)=-\\frac{1}{3}", "steps": [ { "type": "step", "primary": "Let: $$\\sec\\left(x\\right)=u$$", "result": "-2-\\frac{1}{u}+3u=0" }, { "type": "interim", "title": "$$-2-\\frac{1}{u}+3u=0{\\quad:\\quad}u=1,\\:u=-\\frac{1}{3}$$", "input": "-2-\\frac{1}{u}+3u=0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$u$$", "input": "-2-\\frac{1}{u}+3u=0", "result": "-2u-1+3u^{2}=0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$u$$", "result": "-2u-\\frac{1}{u}u+3uu=0\\cdot\\:u" }, { "type": "interim", "title": "Simplify", "input": "-2u-\\frac{1}{u}u+3uu=0\\cdot\\:u", "result": "-2u-1+3u^{2}=0", "steps": [ { "type": "interim", "title": "Simplify $$-\\frac{1}{u}u:{\\quad}-1$$", "input": "-\\frac{1}{u}u", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{1\\cdot\\:u}{u}" }, { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Ltqef1xqfoqL4fud3pFcWwCWKUbvV6WK3fDUgFtg3Q/OVFlWs+2z46G1OZznvlUaZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0nUvIY1ZuxLrV7/923vrAOsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "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": "-2u-1+3u^{2}=0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve $$-2u-1+3u^{2}=0:{\\quad}u=1,\\:u=-\\frac{1}{3}$$", "input": "-2u-1+3u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "3u^{2}-2u-1=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "3u^{2}-2u-1=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:3\\left(-1\\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 $${\\quad}a=3,\\:b=-2,\\:c=-1$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:3\\left(-1\\right)}}{2\\cdot\\:3}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bPMyneU6LXHviR2M5I4mDC5OIvHfHjJ9HVZnsQB4uvio/JI5bBgpgExN510TA5cywo6iOxOQeL70yTMBl67ndVrsdW4QlnVHZaYhaD1DKw/yHO39OaqAj2uw1AILHkpQMV4ZDWvaY95NMqIp7bYciYlpQsukGuuJNMOYOFsnFB2u+em1s4ekk3QV2gkxKLhBNNhkdR6vHMFupVCWJHd0BE+hPmfpWBX0agJ0NurA75wsLWL/oGD++Wi02XE+yuCd0ZyXzQUCjoXvivWM6dCYTeL2uKOmWULehCuWXItXeeiuqTiGbMeMyhyJL8u4mieEbOUTHv9bP8DNxKO6m7IeGiqQAfnwm4hDEAflsdF4Ukl4eAiLOf9ZMs3/+mCbeAn1lAK7ZBoWQ1/FHpVCK6wMj/EqyhRW2Vgp0dJqPBUClzh5HMze/KerEkoiDeO0X1kJj51HCRqdlolTDJhJwLVMLO+VSz2yw3ga51Ab5U4/cafK3wsVCblSmxuxnmJ9OSEUZGkgmoxoNlYnAOkblJbByx3tkbpEhAhHqWbUSSXvVLtFKk3fejFkyiOiq9iG9IkAkK4le0f/h2/I97ZbNwAJwgmBRhN5Vwyd4eFtz3EYGOKMAyxYyd6dEqfPuwJ4V8nZ" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:3\\left(-1\\right)}=4$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:3\\left(-1\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:4}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-2\\right)^{2}+4\\cdot\\:3\\cdot\\:1}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=\\sqrt{2^{2}+4\\cdot\\:3\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3\\cdot\\:1=12$$", "result": "=\\sqrt{2^{2}+12}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+12}" }, { "type": "step", "primary": "Add the numbers: $$4+12=16$$", "result": "=\\sqrt{16}" }, { "type": "step", "primary": "Factor the number: $$16=4^{2}$$", "result": "=\\sqrt{4^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{4^{2}}=4$$" ], "result": "=4", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTivWPy+wTTvjI1RVxxQ3C40AlilG71elit3w1IBbYN0Pdon1OhrviX+zjeY8LDixH6N6Hv6MoTMtvtU0IQwXdn+yrEFCkHLDWs6zaAkN1X/8V+jt3QFVSi8V6RGBH3474iS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-2\\right)+4}{2\\cdot\\:3},\\:{u}_{2}=\\frac{-\\left(-2\\right)-4}{2\\cdot\\:3}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)+4}{2\\cdot\\:3}:{\\quad}1$$", "input": "\\frac{-\\left(-2\\right)+4}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2+4}{2\\cdot\\:3}" }, { "type": "step", "primary": "Add the numbers: $$2+4=6$$", "result": "=\\frac{6}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{6}{6}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kys+PIUFzWnZUwnQQgWTNje1G/V9OcWdVL7fszKOecUgJ/ZZA32ZInFBpDtxBfiK7J5E5gGi2xwchkRMjoVJ7ir6EdYdh/n2c4DPMiuUGOJj8C/f6PsaUffGc0wlKYubNfoWTkn02OZVZw64genrbg==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)-4}{2\\cdot\\:3}:{\\quad}-\\frac{1}{3}$$", "input": "\\frac{-\\left(-2\\right)-4}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2-4}{2\\cdot\\:3}" }, { "type": "step", "primary": "Subtract the numbers: $$2-4=-2$$", "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75nsEWDqowPpIr7OS5lECwTe1G/V9OcWdVL7fszKOecUgJ/ZZA32ZInFBpDtxBfiKmWiTEpQjat3SO7/l2m58l81j2ZZIW8Tm92w7y1ZxW5PFyoUUXuvn7kSsC/X/60sUGt71uwR1UFPSiOXfhbvqM1dl41TRHpIfCbRt2Vw5U0E=" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=1,\\:u=-\\frac{1}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=1,\\:u=-\\frac{1}{3}" }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}u=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$-2-\\frac{1}{u}+3u$$ 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=1,\\:u=-\\frac{1}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(x\\right)$$", "result": "\\sec\\left(x\\right)=1,\\:\\sec\\left(x\\right)=-\\frac{1}{3}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sec\\left(x\\right)=1{\\quad:\\quad}x=2πn$$", "input": "\\sec\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sec\\left(x\\right)=1$$", "result": "x=0+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=0+2πn" } ], "meta": { "interimType": "Trig General Solutions sec 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn" } ], "meta": { "interimType": "N/A" } }, { "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": "step", "primary": "Combine all the solutions", "result": "x=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)-\\cos(x)-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }