(tan(x)-sec(x))^2=3

{ "query": { "display": "$$\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}=3$$", "symbolab_question": "EQUATION#(\\tan(x)-\\sec(x))^{2}=3" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{7π}{6}+2πn,x=\\frac{11π}{6}+2πn", "degrees": "x=210^{\\circ }+360^{\\circ }n,x=330^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}=3{\\quad:\\quad}x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}=3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}-3=0" }, { "type": "step", "primary": "Express with sin, cos", "result": "\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}-\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}-3=0" }, { "type": "interim", "title": "Simplify $$\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}-\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}-3:{\\quad}\\frac{\\left(\\sin\\left(x\\right)-1\\right)^{2}-3\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "input": "\\left(\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}-\\frac{1}{\\cos\\left(x\\right)}\\right)^{2}-3", "steps": [ { "type": "interim", "title": "Combine the fractions $$\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}-\\frac{1}{\\cos\\left(x\\right)}:{\\quad}\\frac{\\sin\\left(x\\right)-1}{\\cos\\left(x\\right)}$$", "result": "=\\left(\\frac{\\sin\\left(x\\right)-1}{\\cos\\left(x\\right)}\\right)^{2}-3", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\sin\\left(x\\right)-1}{\\cos\\left(x\\right)}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(\\sin\\left(x\\right)-1\\right)^{2}}{\\cos^{2}\\left(x\\right)}-3", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Convert element to fraction: $$3=\\frac{3\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}$$", "result": "=\\frac{\\left(\\sin\\left(x\\right)-1\\right)^{2}}{\\cos^{2}\\left(x\\right)}-\\frac{3\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\left(\\sin\\left(x\\right)-1\\right)^{2}-3\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{\\left(\\sin\\left(x\\right)-1\\right)^{2}-3\\cos^{2}\\left(x\\right)}{\\cos^{2}\\left(x\\right)}=0" }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "\\left(\\sin\\left(x\\right)-1\\right)^{2}-3\\cos^{2}\\left(x\\right)=0" }, { "type": "step", "primary": "Add $$3\\cos^{2}\\left(x\\right)$$ to both sides", "result": "\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1=3\\cos^{2}\\left(x\\right)" }, { "type": "step", "primary": "Square both sides", "result": "\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}=\\left(3\\cos^{2}\\left(x\\right)\\right)^{2}" }, { "type": "step", "primary": "Subtract $$\\left(3\\cos^{2}\\left(x\\right)\\right)^{2}$$ from both sides", "result": "\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-9\\cos^{4}\\left(x\\right)=0" }, { "type": "interim", "title": "Factor $$\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-9\\cos^{4}\\left(x\\right):{\\quad}\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1+3\\cos^{2}\\left(x\\right)\\right)\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1-3\\cos^{2}\\left(x\\right)\\right)$$", "input": "\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-9\\cos^{4}\\left(x\\right)", "steps": [ { "type": "interim", "title": "Rewrite $$\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-9\\cos^{4}\\left(x\\right)$$ as $$\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-\\left(3\\cos^{2}\\left(x\\right)\\right)^{2}$$", "input": "\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-9\\cos^{4}\\left(x\\right)", "result": "=\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-\\left(3\\cos^{2}\\left(x\\right)\\right)^{2}", "steps": [ { "type": "step", "primary": "Rewrite $$9$$ as $$3^{2}$$", "result": "=\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-3^{2}\\cos^{4}\\left(x\\right)" }, { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "secondary": [ "$$\\cos^{4}\\left(x\\right)=\\left(\\cos^{2}\\left(x\\right)\\right)^{2}$$" ], "result": "=\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-3^{2}\\left(\\cos^{2}\\left(x\\right)\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$a^{m}b^{m}=\\left(ab\\right)^{m}$$", "secondary": [ "$$3^{2}\\left(\\cos^{2}\\left(x\\right)\\right)^{2}=\\left(3\\cos^{2}\\left(x\\right)\\right)^{2}$$" ], "result": "=\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-\\left(3\\cos^{2}\\left(x\\right)\\right)^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)^{2}-\\left(3\\cos^{2}\\left(x\\right)\\right)^{2}=\\left(\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)+3\\cos^{2}\\left(x\\right)\\right)\\left(\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)-3\\cos^{2}\\left(x\\right)\\right)$$" ], "result": "=\\left(\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)+3\\cos^{2}\\left(x\\right)\\right)\\left(\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)-3\\cos^{2}\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } }, { "type": "step", "primary": "Refine", "result": "=\\left(\\sin^{2}\\left(x\\right)+3\\cos^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)\\left(\\sin^{2}\\left(x\\right)-3\\cos^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1+3\\cos^{2}\\left(x\\right)\\right)\\left(\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1-3\\cos^{2}\\left(x\\right)\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1+3\\cos^{2}\\left(x\\right)=0\\lor\\:\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1-3\\cos^{2}\\left(x\\right)=0" }, { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1+3\\cos^{2}\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1+3\\cos^{2}\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3\\cos^{2}\\left(x\\right)", "result": "4-2\\sin\\left(x\\right)-2\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$" ], "result": "=1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3\\left(1-\\sin^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}-2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+4$$", "input": "1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=-2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+4", "steps": [ { "type": "interim", "title": "Expand $$3\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}3-3\\sin^{2}\\left(x\\right)$$", "input": "3\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3-3\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=3,\\:b=1,\\:c=\\sin^{2}\\left(x\\right)$$" ], "result": "=3\\cdot\\:1-3\\sin^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:1=3$$", "result": "=3-3\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VfxNYx8Dt5NHtHzJr+8feEPKZBnILZTwiutqYLWXIDCjkVi15I8rBefLi4Iyt2wr61IY8k5daseBdRuxB9o6q2RLd2VwIqlBNByF6663syRU6H0nS++8kDqP632fuVHP0ChcBj6f1274q8nUWCYMvOIASZeFjDtawNGt9P21GJY=" } }, { "type": "interim", "title": "Simplify $$1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3-3\\sin^{2}\\left(x\\right):{\\quad}-2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+4$$", "input": "1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3-3\\sin^{2}\\left(x\\right)", "result": "=-2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+4", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3\\sin^{2}\\left(x\\right)+1+3" }, { "type": "step", "primary": "Add similar elements: $$\\sin^{2}\\left(x\\right)-3\\sin^{2}\\left(x\\right)=-2\\sin^{2}\\left(x\\right)$$", "result": "=-2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1+3" }, { "type": "step", "primary": "Add the numbers: $$1+3=4$$", "result": "=-2\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+4" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D3PoKmjdOsX6mFhASrXfYcgVNzlm8idRqqTIzSV/n1vL3+jn80U4KgZNKAKS+J4xICf2WQN9mSJxQaQ7cQX4ioPvF/FzrunHkpMGwZvF8zRklXaIuEE2M5rFWTFMyvo6hF+dL8nykBALXotHM9g9ae5AIz++qluupTlLFEcE9J0KMh9jMBMXk6zZ7tlU7t0HQ7oyINt/hQUY02j+rRPVJOmA3m8h0M8PsnqGscp8ZqWwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Qtgk1HFWlZNSdjr65o4kE9dDwitflwokgSyg5IuZuSvi/oCpENGmjztgHdkg2edRGjN+yhf4ytKtej4PuCQcN+chTlQM7vMOIrScLDKKXqeZ1/y00S8BIROgMJpP+s4VCAKmQZFvLjG+9kc8embfreLwU03p45cea+sHD2I1PK7OTc3wLsE4ExaGOuYeMP081sD7NfhsPe7eDHrmjY0mE0b3po9xgRNoiA3F1PnEeoiPoMa0NbQl3ipWCCdZ+jS3sIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Solve by substitution", "input": "4-2\\sin\\left(x\\right)-2\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=-2,\\:\\sin\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "4-2u-2u^{2}=0" }, { "type": "interim", "title": "$$4-2u-2u^{2}=0{\\quad:\\quad}u=-2,\\:u=1$$", "input": "4-2u-2u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-2u^{2}-2u+4=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-2u^{2}-2u+4=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}}{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 $${\\quad}a=-2,\\:b=-2,\\:c=4$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}}{2\\left(-2\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}=6$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\left(-2\\right)\\cdot\\:4}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:6}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-2\\right)^{2}+4\\cdot\\:2\\cdot\\:4}" }, { "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\\:2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2\\cdot\\:4=32$$", "result": "=\\sqrt{2^{2}+32}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+32}" }, { "type": "step", "primary": "Add the numbers: $$4+32=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTkyNvvFb3YApY88JKBDgydwAlilG71elit3w1IBbYN0POfzEJvy01qFtiDDf1FBy4KN6Hv6MoTMtvtU0IQwXdn9gemVBIzJhCSDpYD1Ky1x2Rtr5bbogjnqwl8ACFn8KGSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-2\\right)+6}{2\\left(-2\\right)},\\:{u}_{2}=\\frac{-\\left(-2\\right)-6}{2\\left(-2\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)+6}{2\\left(-2\\right)}:{\\quad}-2$$", "input": "\\frac{-\\left(-2\\right)+6}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{2+6}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OFqNWcG7GsnnezBHiIMWJNdzTQFcXvE5M6pxvFYiUr51g99dC9fj9sg0EHzBIRDRwJRuKB0+skz49yokD/nm0nXxzR/D3xpyR5yXTZ2YQF1djZf53lsfXPAzKZQTtCk9vzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)-6}{2\\left(-2\\right)}:{\\quad}1$$", "input": "\\frac{-\\left(-2\\right)-6}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{2-6}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$2-6=-4$$", "result": "=\\frac{-4}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{-4}{-4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{4}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72LopWbBMxyRtD0Gow7zUstdzTQFcXvE5M6pxvFYiUr51g99dC9fj9sg0EHzBIRDR+8ZDu8iF4MSewt4yms1lIcXKhRRe6+fuRKwL9f/rSxT0sdj/x4QjVXxb0xKhtjFEJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=-2,\\:u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=-2,\\:\\sin\\left(x\\right)=1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-2{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=-2", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=1{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=1$$", "result": "x=\\frac{π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{2}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1-3\\cos^{2}\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1-3\\cos^{2}\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3\\cos^{2}\\left(x\\right)", "result": "-2-2\\sin\\left(x\\right)+4\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Pythagorean identity: $$\\cos^{2}\\left(x\\right)+\\sin^{2}\\left(x\\right)=1$$", "secondary": [ "$$\\cos^{2}\\left(x\\right)=1-\\sin^{2}\\left(x\\right)$$" ], "result": "=1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3\\left(1-\\sin^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}4\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-2$$", "input": "1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=4\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-2", "steps": [ { "type": "interim", "title": "Expand $$-3\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}-3+3\\sin^{2}\\left(x\\right)$$", "input": "-3\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3+3\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-3,\\:b=1,\\:c=\\sin^{2}\\left(x\\right)$$" ], "result": "=-3\\cdot\\:1-\\left(-3\\right)\\sin^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-3\\cdot\\:1+3\\sin^{2}\\left(x\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:1=3$$", "result": "=-3+3\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7oIbKPaCJ+q5riWsx1FjM9DD8PgXlmKgFspRJo/rFGenMwViaLUXkeD+JukROhWdjLrcx3idLbBKF0swuaCUj8O5byrQDQVCXUD0vH/fvOdz8bYA0b6V2RSTOZ7Os9NOD0C4zIJjMWo2aKrWrHU9i1YLF6drOTclI884E0jj8Wjs=" } }, { "type": "interim", "title": "Simplify $$1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3+3\\sin^{2}\\left(x\\right):{\\quad}4\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-2$$", "input": "1+\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-3+3\\sin^{2}\\left(x\\right)", "result": "=4\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-2", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+3\\sin^{2}\\left(x\\right)+1-3" }, { "type": "step", "primary": "Add similar elements: $$\\sin^{2}\\left(x\\right)+3\\sin^{2}\\left(x\\right)=4\\sin^{2}\\left(x\\right)$$", "result": "=4\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)+1-3" }, { "type": "step", "primary": "Add/Subtract the numbers: $$1-3=-2$$", "result": "=4\\sin^{2}\\left(x\\right)-2\\sin\\left(x\\right)-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D3PoKmjdOsX6mFhASrXfYVcYOv+PvRfDcomAQxokmDDL3+jn80U4KgZNKAKS+J4xICf2WQN9mSJxQaQ7cQX4iuVI9vQqA0lid82zm0v1gzvzYk6/N+LKmRocaCAaplF8jqKbenwZGuBz+HX+SewUnR4pgUWEah0lniZLlD4X0wvaaHsoD7NJBIU/DQVHipg7uqL0TlvWhmm9zRBDdwCFQAoJStNECeOvPhjMz+qHGVG/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Qtgk1HFWlZNSdjr65o4kE9dDwitflwokgSyg5IuZuSuOByTscB4SvF9JK4eUZekpGjN+yhf4ytKtej4PuCQcN+chTlQM7vMOIrScLDKKXqeZ1/y00S8BIROgMJpP+s4VCAKmQZFvLjG+9kc8embfrcjXhMAtGeoXFql50XaKCvSR6OLSieQGhuieRyuQd8mEeqXxdc+rps1CUyb7fqI2GRnjzbYY5XzfRclpBEEnti9RB3DPW8+sYW5R2mkRkguQvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "interim", "title": "Solve by substitution", "input": "-2-2\\sin\\left(x\\right)+4\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=1,\\:\\sin\\left(x\\right)=-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "-2-2u+4u^{2}=0" }, { "type": "interim", "title": "$$-2-2u+4u^{2}=0{\\quad:\\quad}u=1,\\:u=-\\frac{1}{2}$$", "input": "-2-2u+4u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "4u^{2}-2u-2=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "4u^{2}-2u-2=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}}{2\\cdot\\:4}", "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 $${\\quad}a=4,\\:b=-2,\\:c=-2$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}}{2\\cdot\\:4}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}=6$$", "input": "\\sqrt{\\left(-2\\right)^{2}-4\\cdot\\:4\\left(-2\\right)}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-2\\right)\\pm\\:6}{2\\cdot\\:4}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{\\left(-2\\right)^{2}+4\\cdot\\:4\\cdot\\:2}" }, { "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\\:4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:4\\cdot\\:2=32$$", "result": "=\\sqrt{2^{2}+32}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+32}" }, { "type": "step", "primary": "Add the numbers: $$4+32=36$$", "result": "=\\sqrt{36}" }, { "type": "step", "primary": "Factor the number: $$36=6^{2}$$", "result": "=\\sqrt{6^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{6^{2}}=6$$" ], "result": "=6", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7z3fhNomdwT8bNJhnZYfuTsxHTuOb6t4zxqqdK8Vq2s0AlilG71elit3w1IBbYN0POfzEJvy01qFtiDDf1FBy4KN6Hv6MoTMtvtU0IQwXdn+yrEFCkHLDWs6zaAkN1X/8qrd2ABMYpEWKX4ChVYuxQSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-2\\right)+6}{2\\cdot\\:4},\\:{u}_{2}=\\frac{-\\left(-2\\right)-6}{2\\cdot\\:4}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)+6}{2\\cdot\\:4}:{\\quad}1$$", "input": "\\frac{-\\left(-2\\right)+6}{2\\cdot\\:4}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2+6}{2\\cdot\\:4}" }, { "type": "step", "primary": "Add the numbers: $$2+6=8$$", "result": "=\\frac{8}{2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{8}{8}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OFqNWcG7GsnnezBHiIMWJM7MSFto1NUhp9pQLIi+CBIgJ/ZZA32ZInFBpDtxBfiK7J5E5gGi2xwchkRMjoVJ7ir6EdYdh/n2c4DPMiuUGOKjs3X5Ur8lPZHo6jPummXmDkVdUy3FPlmBP7wKv/Ym5g==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-2\\right)-6}{2\\cdot\\:4}:{\\quad}-\\frac{1}{2}$$", "input": "\\frac{-\\left(-2\\right)-6}{2\\cdot\\:4}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{2-6}{2\\cdot\\:4}" }, { "type": "step", "primary": "Subtract the numbers: $$2-6=-4$$", "result": "=\\frac{-4}{2\\cdot\\:4}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:4=8$$", "result": "=\\frac{-4}{8}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{4}{8}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s72LopWbBMxyRtD0Gow7zUss7MSFto1NUhp9pQLIi+CBIgJ/ZZA32ZInFBpDtxBfiKmWiTEpQjat3SO7/l2m58l6mNsLv4gXj0VPO6vgHds8nFyoUUXuvn7kSsC/X/60sUg8++KpDen8wQEBKt2w2vdrd5kDP7mPhrbng2c5V4Fjk=" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=1,\\:u=-\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=1,\\:\\sin\\left(x\\right)=-\\frac{1}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=1{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=1$$", "result": "x=\\frac{π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\frac{1}{2}{\\quad:\\quad}x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn$$", "input": "\\sin\\left(x\\right)=-\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-\\frac{1}{2}$$", "result": "x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" }, { "type": "interim", "title": "Verify solutions by plugging them into the original equation", "steps": [ { "type": "step", "primary": "Check the solutions by plugging them into $$\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}=3$$<br/>Remove the ones that don't agree with the equation." }, { "type": "interim", "title": "Check the solution $$\\frac{π}{2}+2πn:{\\quad}$$False", "input": "\\frac{π}{2}+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\frac{π}{2}+2π1" }, { "type": "step", "primary": "For $$\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}=3{\\quad}$$plug in$${\\quad}x=\\frac{π}{2}+2π1$$", "result": "\\left(\\tan\\left(\\frac{π}{2}+2π1\\right)-\\sec\\left(\\frac{π}{2}+2π1\\right)\\right)^{2}=3" }, { "type": "step", "result": "\\mathrm{Undefined}" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{False}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$\\frac{7π}{6}+2πn:{\\quad}$$True", "input": "\\frac{7π}{6}+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\frac{7π}{6}+2π1" }, { "type": "step", "primary": "For $$\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}=3{\\quad}$$plug in$${\\quad}x=\\frac{7π}{6}+2π1$$", "result": "\\left(\\tan\\left(\\frac{7π}{6}+2π1\\right)-\\sec\\left(\\frac{7π}{6}+2π1\\right)\\right)^{2}=3" }, { "type": "step", "primary": "Refine", "result": "3=3" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } }, { "type": "interim", "title": "Check the solution $$\\frac{11π}{6}+2πn:{\\quad}$$True", "input": "\\frac{11π}{6}+2πn", "steps": [ { "type": "step", "primary": "Plug in $$n=1$$", "result": "\\frac{11π}{6}+2π1" }, { "type": "step", "primary": "For $$\\left(\\tan\\left(x\\right)-\\sec\\left(x\\right)\\right)^{2}=3{\\quad}$$plug in$${\\quad}x=\\frac{11π}{6}+2π1$$", "result": "\\left(\\tan\\left(\\frac{11π}{6}+2π1\\right)-\\sec\\left(\\frac{11π}{6}+2π1\\right)\\right)^{2}=3" }, { "type": "step", "primary": "Refine", "result": "3=3" }, { "type": "step", "result": "\\Rightarrow\\:\\mathrm{True}" } ], "meta": { "interimType": "Check One Solution 1Eq" } } ], "meta": { "interimType": "Check Solutions Plug Preface 1Eq" } }, { "type": "step", "result": "x=\\frac{7π}{6}+2πn,\\:x=\\frac{11π}{6}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "(\\tan(x)-\\sec(x))^{2}-3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }