10sin^2(2u)+6cos^2(2u)=8

{ "query": { "display": "$$10\\sin^{2}\\left(2u\\right)+6\\cos^{2}\\left(2u\\right)=8$$", "symbolab_question": "EQUATION#10\\sin^{2}(2u)+6\\cos^{2}(2u)=8" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "u=\\frac{π}{8}+πn,u=\\frac{3π}{8}+πn,u=\\frac{5π}{8}+πn,u=\\frac{7π}{8}+πn", "degrees": "u=22.5^{\\circ }+180^{\\circ }n,u=67.5^{\\circ }+180^{\\circ }n,u=112.5^{\\circ }+180^{\\circ }n,u=157.5^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$10\\sin^{2}\\left(2u\\right)+6\\cos^{2}\\left(2u\\right)=8{\\quad:\\quad}u=\\frac{π}{8}+πn,\\:u=\\frac{3π}{8}+πn,\\:u=\\frac{5π}{8}+πn,\\:u=\\frac{7π}{8}+πn$$", "input": "10\\sin^{2}\\left(2u\\right)+6\\cos^{2}\\left(2u\\right)=8", "steps": [ { "type": "step", "primary": "Subtract $$8$$ from both sides", "result": "10\\sin^{2}\\left(2u\\right)+6\\cos^{2}\\left(2u\\right)-8=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-8+10\\sin^{2}\\left(2u\\right)+6\\cos^{2}\\left(2u\\right)", "result": "-2+4\\sin^{2}\\left(2u\\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": "=-8+10\\sin^{2}\\left(2u\\right)+6\\left(1-\\sin^{2}\\left(2u\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-8+10\\sin^{2}\\left(2u\\right)+6\\left(1-\\sin^{2}\\left(2u\\right)\\right):{\\quad}4\\sin^{2}\\left(2u\\right)-2$$", "input": "-8+10\\sin^{2}\\left(2u\\right)+6\\left(1-\\sin^{2}\\left(2u\\right)\\right)", "result": "=4\\sin^{2}\\left(2u\\right)-2", "steps": [ { "type": "interim", "title": "Expand $$6\\left(1-\\sin^{2}\\left(2u\\right)\\right):{\\quad}6-6\\sin^{2}\\left(2u\\right)$$", "input": "6\\left(1-\\sin^{2}\\left(2u\\right)\\right)", "result": "=-8+10\\sin^{2}\\left(2u\\right)+6-6\\sin^{2}\\left(2u\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=6,\\:b=1,\\:c=\\sin^{2}\\left(2u\\right)$$" ], "result": "=6\\cdot\\:1-6\\sin^{2}\\left(2u\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$6\\cdot\\:1=6$$", "result": "=6-6\\sin^{2}\\left(2u\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CXSX/DSkK9HXdDOzH5k8Du0SGayEI3RHdjiHWfmmTkXMwViaLUXkeD+JukROhWdjhqkbVDFhOsbUAffX8WvKN+5byrQDQVCXUD0vH/fvOdz8bYA0b6V2RSTOZ7Os9NODTDWsmsSoj0XHPXuukGd3hGeZxoYnuj+rBtmCbtHdCQI=" } }, { "type": "interim", "title": "Simplify $$-8+10\\sin^{2}\\left(2u\\right)+6-6\\sin^{2}\\left(2u\\right):{\\quad}4\\sin^{2}\\left(2u\\right)-2$$", "input": "-8+10\\sin^{2}\\left(2u\\right)+6-6\\sin^{2}\\left(2u\\right)", "result": "=4\\sin^{2}\\left(2u\\right)-2", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=10\\sin^{2}\\left(2u\\right)-6\\sin^{2}\\left(2u\\right)-8+6" }, { "type": "step", "primary": "Add similar elements: $$10\\sin^{2}\\left(2u\\right)-6\\sin^{2}\\left(2u\\right)=4\\sin^{2}\\left(2u\\right)$$", "result": "=4\\sin^{2}\\left(2u\\right)-8+6" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-8+6=-2$$", "result": "=4\\sin^{2}\\left(2u\\right)-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s779S9WCkmpNbhwMFEr4cYZ6n3RZmSBvZvU5ZouKv+koP6qG/pOBFcKnL9mJ2KCclECUCWbkwGOY7PqKo3U/JLJXEYsK9iFetWEqU7YHi8gmH3HeB6+Z6WHxiYcwBuJWGkmNwEgvmmajzdQwVlFLMmxVAD52N4jrwcvQR9QaG+4iBZbapRqJmqGLQ+yUkLCpZw6e8HQH+WAPTEGDUMCMuiLg==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qQc06XVF4upWITwZ65f8vYyjb1dP0NcyfGscOjyepZE2nEhcHIFFTVYYB+9XrwfcgWbCoew6kQ2Oq25Lg2vIzGnPkt49vEcA7X0gWTEYldq3XZSyKuftMLhEd03LjxEbiO+PbCpEt1pIB4gZrBOTJ+CaYeqlh3sro/2OupNOPErwt9LEn7QCBUukJKctfSJKk2VQdqPGsmYTaYyesllHTRc4KXs8xgJNiLkJwr0b/5OJqVxX90jlMfh9fKn6dzC4" } }, { "type": "interim", "title": "Solve by substitution", "input": "-2+4\\sin^{2}\\left(2u\\right)=0", "result": "\\sin\\left(2u\\right)=\\sqrt{\\frac{1}{2}},\\:\\sin\\left(2u\\right)=-\\sqrt{\\frac{1}{2}}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(2u\\right)=v$$", "result": "-2+4v^{2}=0" }, { "type": "interim", "title": "$$-2+4v^{2}=0{\\quad:\\quad}v=\\sqrt{\\frac{1}{2}},\\:v=-\\sqrt{\\frac{1}{2}}$$", "input": "-2+4v^{2}=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "-2+4v^{2}=0", "result": "4v^{2}=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "-2+4v^{2}+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "4v^{2}=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$4$$", "input": "4v^{2}=2", "result": "v^{2}=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$4$$", "result": "\\frac{4v^{2}}{4}=\\frac{2}{4}" }, { "type": "step", "primary": "Simplify", "result": "v^{2}=\\frac{1}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "v=\\sqrt{\\frac{1}{2}},\\:v=-\\sqrt{\\frac{1}{2}}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$v=\\sin\\left(2u\\right)$$", "result": "\\sin\\left(2u\\right)=\\sqrt{\\frac{1}{2}},\\:\\sin\\left(2u\\right)=-\\sqrt{\\frac{1}{2}}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(2u\\right)=\\sqrt{\\frac{1}{2}}{\\quad:\\quad}u=\\frac{π}{8}+πn,\\:u=\\frac{3π}{8}+πn$$", "input": "\\sin\\left(2u\\right)=\\sqrt{\\frac{1}{2}}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2u\\right)=\\sqrt{\\frac{1}{2}}$$", "result": "2u=\\frac{π}{4}+2πn,\\:2u=\\frac{3π}{4}+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": "2u=\\frac{π}{4}+2πn,\\:2u=\\frac{3π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2u=\\frac{π}{4}+2πn:{\\quad}u=\\frac{π}{8}+πn$$", "input": "2u=\\frac{π}{4}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2u=\\frac{π}{4}+2πn", "result": "u=\\frac{π}{8}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2u}{2}=\\frac{\\frac{π}{4}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2u}{2}=\\frac{\\frac{π}{4}}{2}+\\frac{2πn}{2}", "result": "u=\\frac{π}{8}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2u}{2}:{\\quad}u$$", "input": "\\frac{2u}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=u" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gsv62VfcM1CB0UUPgOajeS061ljBSPJeENOw2efoSWt0V0+5rHisr1JlO5pdMn7MRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oBF2YvZdxUtLndzkaBJSF6" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{4}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{π}{8}+πn$$", "input": "\\frac{\\frac{π}{4}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{4}}{2}=\\frac{π}{8}$$", "input": "\\frac{\\frac{π}{4}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=\\frac{π}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajfBUDCK2p4efPsnQUA32SOjNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66G76DMDH258ED6TCmtXj8Vmq+/XUfFY44oQy7yrU6sldleyoAWyEiMC09UXmm0ysSzygo1WVqYqDorj9mXvCvqes=" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{π}{8}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXnqeQkVqfwvjiIs6Ts5NPzCl1oDQxTLS01lmG4qDSnCzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhuRVSaXRzps41ft/HVLsAQ8eqXxdc+rps1CUyb7fqI2GWOC1RGw1nyYXaraVAOvzodb3g/IJJ/7smJgVXcZNhHFMraKqGjoVZ5Ymz5GNk+xPPotFMd5aa71+dAhX5ea6wE=" } }, { "type": "step", "result": "u=\\frac{π}{8}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2u=\\frac{3π}{4}+2πn:{\\quad}u=\\frac{3π}{8}+πn$$", "input": "2u=\\frac{3π}{4}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2u=\\frac{3π}{4}+2πn", "result": "u=\\frac{3π}{8}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2u}{2}=\\frac{\\frac{3π}{4}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2u}{2}=\\frac{\\frac{3π}{4}}{2}+\\frac{2πn}{2}", "result": "u=\\frac{3π}{8}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2u}{2}:{\\quad}u$$", "input": "\\frac{2u}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=u" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gsv62VfcM1CB0UUPgOajeS061ljBSPJeENOw2efoSWt0V0+5rHisr1JlO5pdMn7MRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oBF2YvZdxUtLndzkaBJSF6" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{3π}{4}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{3π}{8}+πn$$", "input": "\\frac{\\frac{3π}{4}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{3π}{4}}{2}=\\frac{3π}{8}$$", "input": "\\frac{\\frac{3π}{4}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=\\frac{3π}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajb/b0aK6mxzkrRQ3Y2KdbDjdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldRK0Yo9KYWhFtMkDP9QQgDmMWtOub2dAxgaWacwKjDMkB6iXzgR8rXBT7/gCovYjN/VysbmH6V1a/M1Gp5krfeRUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{3π}{8}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajTSexn9aeRxnCqKa8zEDxn22z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXa3zET8py0nz91WHtTkJXURQ+kwmjhUC66VagwwWc+BL8LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJcIdcrEX3CFtz7dlmgYCS4te50ZSd1lOulQrxxkWhKPlIL+85zi6ibrzOxBmpP0LLM=" } }, { "type": "step", "result": "u=\\frac{3π}{8}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=\\frac{π}{8}+πn,\\:u=\\frac{3π}{8}+πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(2u\\right)=-\\sqrt{\\frac{1}{2}}{\\quad:\\quad}u=\\frac{5π}{8}+πn,\\:u=\\frac{7π}{8}+πn$$", "input": "\\sin\\left(2u\\right)=-\\sqrt{\\frac{1}{2}}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2u\\right)=-\\sqrt{\\frac{1}{2}}$$", "result": "2u=\\frac{5π}{4}+2πn,\\:2u=\\frac{7π}{4}+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": "2u=\\frac{5π}{4}+2πn,\\:2u=\\frac{7π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2u=\\frac{5π}{4}+2πn:{\\quad}u=\\frac{5π}{8}+πn$$", "input": "2u=\\frac{5π}{4}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2u=\\frac{5π}{4}+2πn", "result": "u=\\frac{5π}{8}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2u}{2}=\\frac{\\frac{5π}{4}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2u}{2}=\\frac{\\frac{5π}{4}}{2}+\\frac{2πn}{2}", "result": "u=\\frac{5π}{8}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2u}{2}:{\\quad}u$$", "input": "\\frac{2u}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=u" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gsv62VfcM1CB0UUPgOajeS061ljBSPJeENOw2efoSWt0V0+5rHisr1JlO5pdMn7MRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oBF2YvZdxUtLndzkaBJSF6" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{5π}{4}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{5π}{8}+πn$$", "input": "\\frac{\\frac{5π}{4}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{5π}{4}}{2}=\\frac{5π}{8}$$", "input": "\\frac{\\frac{5π}{4}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5π}{4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=\\frac{5π}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajc8Kldky6Rzz/53zsu+qL5Ddd47a0hQ8flDbGsI5To1dYDPq1Ru8svsR9Rij2+7AAK0Yo9KYWhFtMkDP9QQgDmMWtOub2dAxgaWacwKjDMkBvVyFHHVDt5CCyxstIjDRxVnEezVrCeXbY3hwxU1v5wgkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{5π}{8}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajQFYZdnKB67CgZS5jvX7z+e2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXWAz6tUbvLL7EfUYo9vuwABQ+kwmjhUC66VagwwWc+BL8LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJeHPmfvXfbrgKsGcF+8iHDpe50ZSd1lOulQrxxkWhKPlIL+85zi6ibrzOxBmpP0LLM=" } }, { "type": "step", "result": "u=\\frac{5π}{8}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2u=\\frac{7π}{4}+2πn:{\\quad}u=\\frac{7π}{8}+πn$$", "input": "2u=\\frac{7π}{4}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2u=\\frac{7π}{4}+2πn", "result": "u=\\frac{7π}{8}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2u}{2}=\\frac{\\frac{7π}{4}}{2}+\\frac{2πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2u}{2}=\\frac{\\frac{7π}{4}}{2}+\\frac{2πn}{2}", "result": "u=\\frac{7π}{8}+πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2u}{2}:{\\quad}u$$", "input": "\\frac{2u}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=u" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Gsv62VfcM1CB0UUPgOajeS061ljBSPJeENOw2efoSWt0V0+5rHisr1JlO5pdMn7MRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6oBF2YvZdxUtLndzkaBJSF6" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{7π}{4}}{2}+\\frac{2πn}{2}:{\\quad}\\frac{7π}{8}+πn$$", "input": "\\frac{\\frac{7π}{4}}{2}+\\frac{2πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{7π}{4}}{2}=\\frac{7π}{8}$$", "input": "\\frac{\\frac{7π}{4}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{7π}{4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=\\frac{7π}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajfT7os/Cht8hOvhXzhmzQ0Ldd47a0hQ8flDbGsI5To1dVESQA48qouUS12usnKDA2K0Yo9KYWhFtMkDP9QQgDmMWtOub2dAxgaWacwKjDMkBYE5Zs/0jWlAacxKuqfjbzVl2IRVSAaOPX2jdam1+lickt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "$$\\frac{2πn}{2}=πn$$", "input": "\\frac{2πn}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YVyjdFRW0jf8Bxzp61JzqHyRHuGw7+tM5METTDj6vVECnBmppnUV+797UqKeeyBQmGP7cF4dEaUuqswlNsH4qzvxhqyDNGf7EUmy3SORhvXPUvVtHPbCNTQhn1k5UbDm" } }, { "type": "step", "result": "=\\frac{7π}{8}+πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXu9e+g89ZcRl+5GMQGdJ9+2z3vSzoDYIK8LwhuQjcbO3XeO2tIUPH5Q2xrCOU6NXVREkAOPKqLlEtdrrJygwNhQ+kwmjhUC66VagwwWc+BL8LfSxJ+0AgVLpCSnLX0iSpuSlq7+cu05ZlcV/9tnDJfnGHyD65Q7DBtAQTLwzSHqe50ZSd1lOulQrxxkWhKPlIL+85zi6ibrzOxBmpP0LLM=" } }, { "type": "step", "result": "u=\\frac{7π}{8}+πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=\\frac{5π}{8}+πn,\\:u=\\frac{7π}{8}+πn" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "u=\\frac{π}{8}+πn,\\:u=\\frac{3π}{8}+πn,\\:u=\\frac{5π}{8}+πn,\\:u=\\frac{7π}{8}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "u", "plotRequest": "10\\sin^{2}(2u)+6\\cos^{2}(2u)-8" }, "showViewLarger": true } }, "meta": { "showVerify": true } }