sin(θ)-2cos^2(θ)=-1

{ "query": { "display": "$$\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)=-1$$", "symbolab_question": "EQUATION#\\sin(θ)-2\\cos^{2}(θ)=-1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{π}{6}+2πn,θ=\\frac{5π}{6}+2πn,θ=\\frac{3π}{2}+2πn", "degrees": "θ=30^{\\circ }+360^{\\circ }n,θ=150^{\\circ }+360^{\\circ }n,θ=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)=-1{\\quad:\\quad}θ=\\frac{π}{6}+2πn,\\:θ=\\frac{5π}{6}+2πn,\\:θ=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)=-1", "steps": [ { "type": "step", "primary": "Subtract $$-1$$ from both sides", "result": "\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)+1=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "1+\\sin\\left(θ\\right)-2\\cos^{2}\\left(θ\\right)", "result": "-1+\\sin\\left(θ\\right)+2\\sin^{2}\\left(θ\\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\\left(θ\\right)-2\\left(1-\\sin^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$1+\\sin\\left(θ\\right)-2\\left(1-\\sin^{2}\\left(θ\\right)\\right):{\\quad}2\\sin^{2}\\left(θ\\right)+\\sin\\left(θ\\right)-1$$", "input": "1+\\sin\\left(θ\\right)-2\\left(1-\\sin^{2}\\left(θ\\right)\\right)", "result": "=2\\sin^{2}\\left(θ\\right)+\\sin\\left(θ\\right)-1", "steps": [ { "type": "interim", "title": "Expand $$-2\\left(1-\\sin^{2}\\left(θ\\right)\\right):{\\quad}-2+2\\sin^{2}\\left(θ\\right)$$", "input": "-2\\left(1-\\sin^{2}\\left(θ\\right)\\right)", "result": "=1+\\sin\\left(θ\\right)-2+2\\sin^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-2,\\:b=1,\\:c=\\sin^{2}\\left(θ\\right)$$" ], "result": "=-2\\cdot\\:1-\\left(-2\\right)\\sin^{2}\\left(θ\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-2\\cdot\\:1+2\\sin^{2}\\left(θ\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=-2+2\\sin^{2}\\left(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7YhJZK7lFAca7opOSbYFOk61/TqqEqZbGfmhpsXTelQ9wkKGJWEPFPk38sdJMsyPIBoz3N7mGDwCEYDP1CETh/LqcSX+26Db9UvR3NA1yV77e1A9ekahUIWtvNWrN4v94shSVFGFggIVrWo7/+/RB4Vg0PqcanvHRdMKpI42Y7jM=" } }, { "type": "interim", "title": "Simplify $$1+\\sin\\left(θ\\right)-2+2\\sin^{2}\\left(θ\\right):{\\quad}2\\sin^{2}\\left(θ\\right)+\\sin\\left(θ\\right)-1$$", "input": "1+\\sin\\left(θ\\right)-2+2\\sin^{2}\\left(θ\\right)", "result": "=2\\sin^{2}\\left(θ\\right)+\\sin\\left(θ\\right)-1", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\sin\\left(θ\\right)+2\\sin^{2}\\left(θ\\right)+1-2" }, { "type": "step", "primary": "Add/Subtract the numbers: $$1-2=-1$$", "result": "=2\\sin^{2}\\left(θ\\right)+\\sin\\left(θ\\right)-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73/Df/zs+ztBE+AOJVI8I65XrQEegY8fk56vr4B6+mkoAlilG71elit3w1IBbYN0PWpy+Qag6aPu9Qps/I1tnMfyekp/dVxf2LBQnIGm8nJoiJuWkSZHcWa5CRtF54tG1TeQKHeh69S6dnv9vSoUoFMLL6MCNumftSpL57c/nfpjM4aRtGjuxG6QCCfxexDhFvRWS30fgNgDV0ICqvsbNCQ==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Qtgk1HFWlZNSdjr65o4kE2clcabZ4BVHIsE+zeRqsbVw1nH/YQO5vZ1KwvDc0aD3EsLZPw0Vm9MD6l9/kB1c91f2VE3iaVir/MjR2DERcZ0+92eGLSzeVn4MnusSJN6pWhC+mOySu+6iHihPjXMvuIsr1edGuXszzzRHPETmAFVFKk3fejFkyiOiq9iG9IkAO/B7UtNBfw19bDRpX2AlzDEZOhzOLFFCWOVe0TAs9Ps/tVtg7k21KzJIFYRgfW7H" } }, { "type": "interim", "title": "Solve by substitution", "input": "-1+\\sin\\left(θ\\right)+2\\sin^{2}\\left(θ\\right)=0", "result": "\\sin\\left(θ\\right)=\\frac{1}{2},\\:\\sin\\left(θ\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(θ\\right)=u$$", "result": "-1+u+2u^{2}=0" }, { "type": "interim", "title": "$$-1+u+2u^{2}=0{\\quad:\\quad}u=\\frac{1}{2},\\:u=-1$$", "input": "-1+u+2u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "2u^{2}+u-1=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "2u^{2}+u-1=0", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\cdot\\:2\\left(-1\\right)}}{2\\cdot\\:2}", "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 $${\\quad}a=2,\\:b=1,\\:c=-1$$", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\cdot\\:2\\left(-1\\right)}}{2\\cdot\\:2}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{1^{2}-4\\cdot\\:2\\left(-1\\right)}=3$$", "input": "\\sqrt{1^{2}-4\\cdot\\:2\\left(-1\\right)}", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:3}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-4\\cdot\\:2\\left(-1\\right)}" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{1+4\\cdot\\:2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2\\cdot\\:1=8$$", "result": "=\\sqrt{1+8}" }, { "type": "step", "primary": "Add the numbers: $$1+8=9$$", "result": "=\\sqrt{9}" }, { "type": "step", "primary": "Factor the number: $$9=3^{2}$$", "result": "=\\sqrt{3^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{3^{2}}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uac2nmABaurcWf8xH8OQy8yvIR4GfpGQcl34lpvwExnehkKrn0era9rz8TlL+x/vrY9m20n1JUo52Kgu/OA/5rtCR5dIjxQ5ASg+ZPFVSsdKwBTYXZQ/DBdiRybedcI2lfJJYxLxiW6L+sTLeCp4qA==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-1+3}{2\\cdot\\:2},\\:{u}_{2}=\\frac{-1-3}{2\\cdot\\:2}" }, { "type": "interim", "title": "$$u=\\frac{-1+3}{2\\cdot\\:2}:{\\quad}\\frac{1}{2}$$", "input": "\\frac{-1+3}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-1+3=2$$", "result": "=\\frac{2}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{2}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7yk2+y6iN9tn8zyag3vMf9HyPproSfunu4fe9HWJRxzMDnzlbPZjyKgy1eUCFsLd5xXDODtFBCC8Uf836IcE9xxHO0oTnnZveyzJ4AtC1ZGNZ0xxpOdWHqjBkPyzaPf0l9k+wqL9/0DEEyRrZ29aieA==" } }, { "type": "interim", "title": "$$u=\\frac{-1-3}{2\\cdot\\:2}:{\\quad}-1$$", "input": "\\frac{-1-3}{2\\cdot\\:2}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-1-3=-4$$", "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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77KJ7vc3S2KIi8rJV87ccM3yPproSfunu4fe9HWJRxzMDnzlbPZjyKgy1eUCFsLd5pmhpfHqzxU6UVoC7gLWdYz/AeYEIX4I11Ba3Yz03hSDyplju7QZb0IQAGMPjtAFcialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=\\frac{1}{2},\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(θ\\right)$$", "result": "\\sin\\left(θ\\right)=\\frac{1}{2},\\:\\sin\\left(θ\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(θ\\right)=\\frac{1}{2}{\\quad:\\quad}θ=\\frac{π}{6}+2πn,\\:θ=\\frac{5π}{6}+2πn$$", "input": "\\sin\\left(θ\\right)=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(θ\\right)=\\frac{1}{2}$$", "result": "θ=\\frac{π}{6}+2πn,\\:θ=\\frac{5π}{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": "θ=\\frac{π}{6}+2πn,\\:θ=\\frac{5π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(θ\\right)=-1{\\quad:\\quad}θ=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(θ\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(θ\\right)=-1$$", "result": "θ=\\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": "θ=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=\\frac{π}{6}+2πn,\\:θ=\\frac{5π}{6}+2πn,\\:θ=\\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": "θ", "plotRequest": "\\sin(θ)-2\\cos^{2}(θ)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }