41sin^2(x)+32sin(x)-9=0

{ "query": { "display": "$$41\\sin^{2}\\left(x\\right)+32\\sin\\left(x\\right)-9=0$$", "symbolab_question": "EQUATION#41\\sin^{2}(x)+32\\sin(x)-9=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0.22131…+2πn,x=π-0.22131…+2πn,x=\\frac{3π}{2}+2πn", "degrees": "x=12.68038…^{\\circ }+360^{\\circ }n,x=167.31961…^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$41\\sin^{2}\\left(x\\right)+32\\sin\\left(x\\right)-9=0{\\quad:\\quad}x=0.22131…+2πn,\\:x=π-0.22131…+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "41\\sin^{2}\\left(x\\right)+32\\sin\\left(x\\right)-9=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "41\\sin^{2}\\left(x\\right)+32\\sin\\left(x\\right)-9=0", "result": "\\sin\\left(x\\right)=\\frac{9}{41},\\:\\sin\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "41u^{2}+32u-9=0" }, { "type": "interim", "title": "$$41u^{2}+32u-9=0{\\quad:\\quad}u=\\frac{9}{41},\\:u=-1$$", "input": "41u^{2}+32u-9=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "41u^{2}+32u-9=0", "result": "{u}_{1,\\:2}=\\frac{-32\\pm\\:\\sqrt{32^{2}-4\\cdot\\:41\\left(-9\\right)}}{2\\cdot\\:41}", "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 $${\\quad}a=41,\\:b=32,\\:c=-9$$", "result": "{u}_{1,\\:2}=\\frac{-32\\pm\\:\\sqrt{32^{2}-4\\cdot\\:41\\left(-9\\right)}}{2\\cdot\\:41}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{32^{2}-4\\cdot\\:41\\left(-9\\right)}=50$$", "input": "\\sqrt{32^{2}-4\\cdot\\:41\\left(-9\\right)}", "result": "{u}_{1,\\:2}=\\frac{-32\\pm\\:50}{2\\cdot\\:41}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{32^{2}+4\\cdot\\:41\\cdot\\:9}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:41\\cdot\\:9=1476$$", "result": "=\\sqrt{32^{2}+1476}" }, { "type": "step", "primary": "$$32^{2}=1024$$", "result": "=\\sqrt{1024+1476}" }, { "type": "step", "primary": "Add the numbers: $$1024+1476=2500$$", "result": "=\\sqrt{2500}" }, { "type": "step", "primary": "Factor the number: $$2500=50^{2}$$", "result": "=\\sqrt{50^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{50^{2}}=50$$" ], "result": "=50", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hLRdfs6I/RlHsMTCb13Sp1a0fTuAXvryXKuJssEKySUtOtZYwUjyXhDTsNnn6Elru3nFr99nsC2+D1F2toBBgaN6Hv6MoTMtvtU0IQwXdn+hPrSwjIG1vyGaYiIfY8huTi7W7iwuNKdtrNUzBOAVSSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-32+50}{2\\cdot\\:41},\\:{u}_{2}=\\frac{-32-50}{2\\cdot\\:41}" }, { "type": "interim", "title": "$$u=\\frac{-32+50}{2\\cdot\\:41}:{\\quad}\\frac{9}{41}$$", "input": "\\frac{-32+50}{2\\cdot\\:41}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-32+50=18$$", "result": "=\\frac{18}{2\\cdot\\:41}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:41=82$$", "result": "=\\frac{18}{82}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{9}{41}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7r1zRwB26JzPINwk5SnO1bP/SzOhV+PZjXWVN9q1n6FcgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJ0zQESHFrgj8RLEG0nB9aJ3FyoUUXuvn7kSsC/X/60sUh1dXrbKwGZqXmpTDP7sBgmxlIQ/foA0INO/RetNF7uU=" } }, { "type": "interim", "title": "$$u=\\frac{-32-50}{2\\cdot\\:41}:{\\quad}-1$$", "input": "\\frac{-32-50}{2\\cdot\\:41}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-32-50=-82$$", "result": "=\\frac{-82}{2\\cdot\\:41}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:41=82$$", "result": "=\\frac{-82}{82}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{82}{82}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iOG+ht9O0FeQwta5c4tP/P/SzOhV+PZjXWVN9q1n6FcgJ/ZZA32ZInFBpDtxBfiKSECk6GWu1Zs/UVqyB86KAkfQ7pOlkQMojVPaIwryp7mQxpCNNe/w3mfhz+sP3ztF9yUdb6LvYImMMKvSmfKlAA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{9}{41},\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\frac{9}{41},\\:\\sin\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{9}{41}{\\quad:\\quad}x=\\arcsin\\left(\\frac{9}{41}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{9}{41}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\frac{9}{41}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\frac{9}{41}", "result": "x=\\arcsin\\left(\\frac{9}{41}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{9}{41}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\frac{9}{41}$$", "secondary": [ "$$\\sin\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arcsin\\left(a\\right)+2πn,\\:\\quad\\:x=π-\\arcsin\\left(a\\right)+2πn$$" ], "result": "x=\\arcsin\\left(\\frac{9}{41}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{9}{41}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-1{\\quad:\\quad}x=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-1$$", "result": "x=\\frac{3π}{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{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(\\frac{9}{41}\\right)+2πn,\\:x=π-\\arcsin\\left(\\frac{9}{41}\\right)+2πn,\\:x=\\frac{3π}{2}+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=0.22131…+2πn,\\:x=π-0.22131…+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "41\\sin^{2}(x)+32\\sin(x)-9" }, "showViewLarger": true } }, "meta": { "showVerify": true } }