cos(2x)+5sin^2(x)=3

{ "query": { "display": "$$\\cos\\left(2x\\right)+5\\sin^{2}\\left(x\\right)=3$$", "symbolab_question": "EQUATION#\\cos(2x)+5\\sin^{2}(x)=3" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0.95531…+2πn,x=π-0.95531…+2πn,x=-0.95531…+2πn,x=π+0.95531…+2πn", "degrees": "x=54.73561…^{\\circ }+360^{\\circ }n,x=125.26438…^{\\circ }+360^{\\circ }n,x=-54.73561…^{\\circ }+360^{\\circ }n,x=234.73561…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(2x\\right)+5\\sin^{2}\\left(x\\right)=3{\\quad:\\quad}x=0.95531…+2πn,\\:x=π-0.95531…+2πn,\\:x=-0.95531…+2πn,\\:x=π+0.95531…+2πn$$", "input": "\\cos\\left(2x\\right)+5\\sin^{2}\\left(x\\right)=3", "steps": [ { "type": "step", "primary": "Subtract $$3$$ from both sides", "result": "\\cos\\left(2x\\right)+5\\sin^{2}\\left(x\\right)-3=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-3+\\cos\\left(2x\\right)+5\\sin^{2}\\left(x\\right)", "result": "-2+3\\sin^{2}\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\cos\\left(2x\\right)=1-2\\sin^{2}\\left(x\\right)$$", "result": "=-3+1-2\\sin^{2}\\left(x\\right)+5\\sin^{2}\\left(x\\right)" }, { "type": "interim", "title": "Simplify $$-3+1-2\\sin^{2}\\left(x\\right)+5\\sin^{2}\\left(x\\right):{\\quad}3\\sin^{2}\\left(x\\right)-2$$", "input": "-3+1-2\\sin^{2}\\left(x\\right)+5\\sin^{2}\\left(x\\right)", "result": "=3\\sin^{2}\\left(x\\right)-2", "steps": [ { "type": "step", "primary": "Add similar elements: $$-2\\sin^{2}\\left(x\\right)+5\\sin^{2}\\left(x\\right)=3\\sin^{2}\\left(x\\right)$$", "result": "=-3+1+3\\sin^{2}\\left(x\\right)" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-3+1=-2$$", "result": "=3\\sin^{2}\\left(x\\right)-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FD4n7YTbgmF/VRKWuKn/2/N60/lqkoIIRzNMZa6II5GO6Dd3zpuSnKXNDXc0BYg4geXEFCD58rDFDfybZf16JLAUyvNZAD1R86KkDdy6K1Gjeh7+jKEzLb7VNCEMF3Z/bMzoTd+5nEXVeQoBhpFcIBdhGgcmcdNl+GygLENry3JVs+fY72EUz7U3fpf3T2mb4gBJl4WMO1rA0a30/bUYlg==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7qx4evzhvN44rn55B932hTYXnnz7PgLoWPtQLEttIQijkHgfs04QC8frBEUzW43FrEsLZPw0Vm9MD6l9/kB1c91f2VE3iaVir/MjR2DERcZ0+92eGLSzeVn4MnusSJN6pKU9Zd7oSMpF8SJbHA4+1OcIrriY8n/yZD4BfZBKDQGx2WVG1dxuhtwcvc5+As2ToKO+d89V0ksUsI4xf7WSo4wJYYPwif7On5mZsZiNIHrw=" } }, { "type": "interim", "title": "Solve by substitution", "input": "-2+3\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=\\sqrt{\\frac{2}{3}},\\:\\sin\\left(x\\right)=-\\sqrt{\\frac{2}{3}}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "-2+3u^{2}=0" }, { "type": "interim", "title": "$$-2+3u^{2}=0{\\quad:\\quad}u=\\sqrt{\\frac{2}{3}},\\:u=-\\sqrt{\\frac{2}{3}}$$", "input": "-2+3u^{2}=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "-2+3u^{2}=0", "result": "3u^{2}=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "-2+3u^{2}+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "3u^{2}=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3u^{2}=2", "result": "u^{2}=\\frac{2}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3u^{2}}{3}=\\frac{2}{3}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=\\frac{2}{3}" } ], "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": "u=\\sqrt{\\frac{2}{3}},\\:u=-\\sqrt{\\frac{2}{3}}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\sqrt{\\frac{2}{3}},\\:\\sin\\left(x\\right)=-\\sqrt{\\frac{2}{3}}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\sqrt{\\frac{2}{3}}{\\quad:\\quad}x=\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π-\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=\\sqrt{\\frac{2}{3}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=\\sqrt{\\frac{2}{3}}", "result": "x=\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π-\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=\\sqrt{\\frac{2}{3}}$$", "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(\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π-\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\sqrt{\\frac{2}{3}}{\\quad:\\quad}x=\\arcsin\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn$$", "input": "\\sin\\left(x\\right)=-\\sqrt{\\frac{2}{3}}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\sin\\left(x\\right)=-\\sqrt{\\frac{2}{3}}", "result": "x=\\arcsin\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\sin\\left(x\\right)=-\\sqrt{\\frac{2}{3}}$$", "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(-\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π-\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=\\arcsin\\left(-\\sqrt{\\frac{2}{3}}\\right)+2πn,\\:x=π+\\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=0.95531…+2πn,\\:x=π-0.95531…+2πn,\\:x=-0.95531…+2πn,\\:x=π+0.95531…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos(2x)+5\\sin^{2}(x)-3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }