5sec(x)-4cos(x)=8

{ "query": { "display": "$$5\\sec\\left(x\\right)-4\\cos\\left(x\\right)=8$$", "symbolab_question": "EQUATION#5\\sec(x)-4\\cos(x)=8" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{3}+2πn,x=\\frac{5π}{3}+2πn", "degrees": "x=60^{\\circ }+360^{\\circ }n,x=300^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$5\\sec\\left(x\\right)-4\\cos\\left(x\\right)=8{\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "5\\sec\\left(x\\right)-4\\cos\\left(x\\right)=8", "steps": [ { "type": "step", "primary": "Subtract $$8$$ from both sides", "result": "5\\sec\\left(x\\right)-4\\cos\\left(x\\right)-8=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-8-4\\cos\\left(x\\right)+5\\sec\\left(x\\right)", "result": "-8-\\frac{4}{\\sec\\left(x\\right)}+5\\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": "=-8-4\\cdot\\:\\frac{1}{\\sec\\left(x\\right)}+5\\sec\\left(x\\right)" }, { "type": "interim", "title": "$$4\\cdot\\:\\frac{1}{\\sec\\left(x\\right)}=\\frac{4}{\\sec\\left(x\\right)}$$", "input": "4\\cdot\\:\\frac{1}{\\sec\\left(x\\right)}", "result": "=-8-\\frac{4}{\\sec\\left(x\\right)}+5\\sec\\left(x\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{\\sec\\left(x\\right)}" }, { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=\\frac{4}{\\sec\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV86gTrE5NivJpRVqDuGxbtozehkKrn0era9rz8TlL+x/vBVZ9vx5jzfo/n1rSDQAgps+B5ICTt5GH28kHchGdZ9up5CJeIbXg+QMbMdQRewnFS8PgbRY6lX6tcMqfLeRhC8+DQy55dl6Cq10f09Im2VH3G9GVipzXa0Ao3ElDHhRh" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I9FtOZjsccikje0vecythiLkB8TIPrpBYfv9EflP1JUaM37KF/jK0q16Pg+4JBw35yFOVAzu8w4itJwsMopep5nX/LTRLwEhE6Awmk/6zhUIAqZBkW8uMb72Rzx6Zt+tNiRh+dh83te72SMpmYf+WhL6G1mjeOJ/ofg01g9m4BpkS3dlcCKpQTQcheuut7MkmLRx2SCfDYc+NFWlwczFlshyJhXONoCiURo2RU420EFGmlM2AwWiez1syRPMgeig" } }, { "type": "interim", "title": "Solve by substitution", "input": "-8-\\frac{4}{\\sec\\left(x\\right)}+5\\sec\\left(x\\right)=0", "result": "\\sec\\left(x\\right)=2,\\:\\sec\\left(x\\right)=-\\frac{2}{5}", "steps": [ { "type": "step", "primary": "Let: $$\\sec\\left(x\\right)=u$$", "result": "-8-\\frac{4}{u}+5u=0" }, { "type": "interim", "title": "$$-8-\\frac{4}{u}+5u=0{\\quad:\\quad}u=2,\\:u=-\\frac{2}{5}$$", "input": "-8-\\frac{4}{u}+5u=0", "steps": [ { "type": "interim", "title": "Multiply both sides by $$u$$", "input": "-8-\\frac{4}{u}+5u=0", "result": "-8u-4+5u^{2}=0", "steps": [ { "type": "step", "primary": "Multiply both sides by $$u$$", "result": "-8u-\\frac{4}{u}u+5uu=0\\cdot\\:u" }, { "type": "interim", "title": "Simplify", "input": "-8u-\\frac{4}{u}u+5uu=0\\cdot\\:u", "result": "-8u-4+5u^{2}=0", "steps": [ { "type": "interim", "title": "Simplify $$-\\frac{4}{u}u:{\\quad}-4$$", "input": "-\\frac{4}{u}u", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{4u}{u}" }, { "type": "step", "primary": "Cancel the common factor: $$u$$", "result": "=-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cKjcFe6YGDQbyWBIf4rBBgCWKUbvV6WK3fDUgFtg3Q8rjNTTwq/M5na/Z7u3TlijZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2CsKEzYXDgpHUPuN7Jb+kJsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$5uu:{\\quad}5u^{2}$$", "input": "5uu", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$uu=\\:u^{1+1}$$" ], "result": "=5u^{1+1}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=5u^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7THwcnOt1LyliuPDY17izlXCQoYlYQ8U+Tfyx0kyzI8jFoq1UzyWCHGWM3vTDT7pjeqXxdc+rps1CUyb7fqI2GU+oOQh8gvh3XwtQ/RSg3UQkt3WiGR7ZaCaXvz77bMjS" } }, { "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": "-8u-4+5u^{2}=0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Solve $$-8u-4+5u^{2}=0:{\\quad}u=2,\\:u=-\\frac{2}{5}$$", "input": "-8u-4+5u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "5u^{2}-8u-4=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "5u^{2}-8u-4=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:\\sqrt{\\left(-8\\right)^{2}-4\\cdot\\:5\\left(-4\\right)}}{2\\cdot\\:5}", "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 $${\\quad}a=5,\\:b=-8,\\:c=-4$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:\\sqrt{\\left(-8\\right)^{2}-4\\cdot\\:5\\left(-4\\right)}}{2\\cdot\\:5}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-8\\right)^{2}-4\\cdot\\:5\\left(-4\\right)}=12$$", "input": "\\sqrt{\\left(-8\\right)^{2}-4\\cdot\\:5\\left(-4\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-8\\right)\\pm\\:12}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-8\\right)^{2}+4\\cdot\\:5\\cdot\\:4}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-8\\right)^{2}=8^{2}$$" ], "result": "=\\sqrt{8^{2}+4\\cdot\\:5\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:5\\cdot\\:4=80$$", "result": "=\\sqrt{8^{2}+80}" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=\\sqrt{64+80}" }, { "type": "step", "primary": "Add the numbers: $$64+80=144$$", "result": "=\\sqrt{144}" }, { "type": "step", "primary": "Factor the number: $$144=12^{2}$$", "result": "=\\sqrt{12^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{12^{2}}=12$$" ], "result": "=12", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7cd7Mdcgz1T8BBymvdP1bZF9aUPkngOwnWR595nkBUKYAlilG71elit3w1IBbYN0PkraqL8dwk6w6nsgXuGDhcGRLd2VwIqlBNByF6663syTFhi0tFO7E9cbsAqvdFrdf5VVswTDc3WI1DrkS/ETTObCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-8\\right)+12}{2\\cdot\\:5},\\:{u}_{2}=\\frac{-\\left(-8\\right)-12}{2\\cdot\\:5}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-8\\right)+12}{2\\cdot\\:5}:{\\quad}2$$", "input": "\\frac{-\\left(-8\\right)+12}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{8+12}{2\\cdot\\:5}" }, { "type": "step", "primary": "Add the numbers: $$8+12=20$$", "result": "=\\frac{20}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{20}{10}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{20}{10}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7e4xoRT8wCESiOFOMMQ+D20geTklxmplCZnPVESKIjPfehkKrn0era9rz8TlL+x/vYS2YS91TkQh9pmLGKRo5TUfQ7pOlkQMojVPaIwryp7na/yEF422BrRLvDdZvmgooELV8YYBAEn1TDxvnBI6vUQ==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-8\\right)-12}{2\\cdot\\:5}:{\\quad}-\\frac{2}{5}$$", "input": "\\frac{-\\left(-8\\right)-12}{2\\cdot\\:5}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{8-12}{2\\cdot\\:5}" }, { "type": "step", "primary": "Subtract the numbers: $$8-12=-4$$", "result": "=\\frac{-4}{2\\cdot\\:5}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:5=10$$", "result": "=\\frac{-4}{10}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{10}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{2}{5}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75o8AVGA7tNYx1BfA+q+YSEgeTklxmplCZnPVESKIjPfehkKrn0era9rz8TlL+x/vttdvQxZI3PlVepHWO3+UghA+px1XBQoQxsVqbvtfqb118c0fw98ackecl02dmEBd8f6ibwBMAcooiGAyhqo6yAC1SQl7hLzfVZAuSlcDv2M=" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=2,\\:u=-\\frac{2}{5}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=2,\\:u=-\\frac{2}{5}" }, { "type": "step", "primary": "Verify Solutions" }, { "type": "interim", "title": "Find undefined (singularity) points:$${\\quad}u=0$$", "steps": [ { "type": "step", "primary": "Take the denominator(s) of $$-8-\\frac{4}{u}+5u$$ 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=2,\\:u=-\\frac{2}{5}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sec\\left(x\\right)$$", "result": "\\sec\\left(x\\right)=2,\\:\\sec\\left(x\\right)=-\\frac{2}{5}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sec\\left(x\\right)=2{\\quad:\\quad}x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn$$", "input": "\\sec\\left(x\\right)=2", "steps": [ { "type": "interim", "title": "General solutions for $$\\sec\\left(x\\right)=2$$", "result": "x=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{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{π}{3}+2πn,\\:x=\\frac{5π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions sec 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sec\\left(x\\right)=-\\frac{2}{5}{\\quad:\\quad}$$No Solution", "input": "\\sec\\left(x\\right)=-\\frac{2}{5}", "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=\\frac{π}{3}+2πn,\\:x=\\frac{5π}{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": "5\\sec(x)-4\\cos(x)-8" }, "showViewLarger": true } }, "meta": { "showVerify": true } }