sin^2(x)+2cos^2(x)=0

{ "query": { "display": "$$\\sin^{2}\\left(x\\right)+2\\cos^{2}\\left(x\\right)=0$$", "symbolab_question": "EQUATION#\\sin^{2}(x)+2\\cos^{2}(x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)+2\\cos^{2}\\left(x\\right)=0{\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\sin^{2}\\left(x\\right)+2\\cos^{2}\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin^{2}\\left(x\\right)+2\\cos^{2}\\left(x\\right)", "result": "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": "=\\sin^{2}\\left(x\\right)+2\\left(1-\\sin^{2}\\left(x\\right)\\right)" }, { "type": "interim", "title": "Simplify $$\\sin^{2}\\left(x\\right)+2\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}-\\sin^{2}\\left(x\\right)+2$$", "input": "\\sin^{2}\\left(x\\right)+2\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=-\\sin^{2}\\left(x\\right)+2", "steps": [ { "type": "interim", "title": "Expand $$2\\left(1-\\sin^{2}\\left(x\\right)\\right):{\\quad}2-2\\sin^{2}\\left(x\\right)$$", "input": "2\\left(1-\\sin^{2}\\left(x\\right)\\right)", "result": "=\\sin^{2}\\left(x\\right)+2-2\\sin^{2}\\left(x\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=2,\\:b=1,\\:c=\\sin^{2}\\left(x\\right)$$" ], "result": "=2\\cdot\\:1-2\\sin^{2}\\left(x\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2-2\\sin^{2}\\left(x\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hukZBvFwZs1yT1SZ13MdEUPKZBnILZTwiutqYLWXIDCjkVi15I8rBefLi4Iyt2wr/IarMn8h2VrKt6QORRY2J2RLd2VwIqlBNByF6663syRU6H0nS++8kDqP632fuVHPJzkFroAPN5RzvSgm6MdXFeIASZeFjDtawNGt9P21GJY=" } }, { "type": "interim", "title": "Simplify $$\\sin^{2}\\left(x\\right)+2-2\\sin^{2}\\left(x\\right):{\\quad}-\\sin^{2}\\left(x\\right)+2$$", "input": "\\sin^{2}\\left(x\\right)+2-2\\sin^{2}\\left(x\\right)", "result": "=-\\sin^{2}\\left(x\\right)+2", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\sin^{2}\\left(x\\right)-2\\sin^{2}\\left(x\\right)+2" }, { "type": "step", "primary": "Add similar elements: $$\\sin^{2}\\left(x\\right)-2\\sin^{2}\\left(x\\right)=-\\sin^{2}\\left(x\\right)$$", "result": "=-\\sin^{2}\\left(x\\right)+2" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yLNZXTvuE0HRAE4mh3rn1emA3m8h0M8PsnqGscp8ZqUAlilG71elit3w1IBbYN0PlbtMv58phRb2qRbzdA+fvHr6OQrTdcuZWnuZRv9c0QLvbBmbuQNTF0TphKZ8Ruva8pOMITYik9N8AtIc49Ww1V87aK55+XeIrEiJ+N+wkNN1VhmS9sryXIBjHbcr4PrE" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7AlY6PORzP4fzAhs5rb1Nz92rATjZmKAkvRdDL8VQmX3kHgfs04QC8frBEUzW43FrEsLZPw0Vm9MD6l9/kB1c91f2VE3iaVir/MjR2DERcZ0+92eGLSzeVn4MnusSJN6pLF4t72cdfZHGHwo3aD/DTa1Tz2ncr2EqaqmQhIhlA+t2WVG1dxuhtwcvc5+As2ToKO+d89V0ksUsI4xf7WSo4wJYYPwif7On5mZsZiNIHrw=" } }, { "type": "interim", "title": "Solve by substitution", "input": "2-\\sin^{2}\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=\\sqrt{2},\\:\\sin\\left(x\\right)=-\\sqrt{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "2-u^{2}=0" }, { "type": "interim", "title": "$$2-u^{2}=0{\\quad:\\quad}u=\\sqrt{2},\\:u=-\\sqrt{2}$$", "input": "2-u^{2}=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "2-u^{2}=0", "result": "-u^{2}=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "2-u^{2}-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "-u^{2}=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-u^{2}=-2", "result": "u^{2}=2", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-u^{2}}{-1}=\\frac{-2}{-1}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=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": "u=\\sqrt{2},\\:u=-\\sqrt{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=\\sqrt{2},\\:\\sin\\left(x\\right)=-\\sqrt{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\sqrt{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=\\sqrt{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\sqrt{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=-\\sqrt{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin^{2}(x)+2\\cos^{2}(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }