2cos^2(θ)+sin(θ)=2

{ "query": { "display": "$$2\\cos^{2}\\left(θ\\right)+\\sin\\left(θ\\right)=2$$", "symbolab_question": "EQUATION#2\\cos^{2}(θ)+\\sin(θ)=2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=2πn,θ=π+2πn,θ=\\frac{π}{6}+2πn,θ=\\frac{5π}{6}+2πn", "degrees": "θ=0^{\\circ }+360^{\\circ }n,θ=180^{\\circ }+360^{\\circ }n,θ=30^{\\circ }+360^{\\circ }n,θ=150^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos^{2}\\left(θ\\right)+\\sin\\left(θ\\right)=2{\\quad:\\quad}θ=2πn,\\:θ=π+2πn,\\:θ=\\frac{π}{6}+2πn,\\:θ=\\frac{5π}{6}+2πn$$", "input": "2\\cos^{2}\\left(θ\\right)+\\sin\\left(θ\\right)=2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "2\\cos^{2}\\left(θ\\right)+\\sin\\left(θ\\right)-2=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-2+\\sin\\left(θ\\right)+2\\cos^{2}\\left(θ\\right)", "result": "\\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": "=-2+\\sin\\left(θ\\right)+2\\left(1-\\sin^{2}\\left(θ\\right)\\right)" }, { "type": "interim", "title": "Simplify $$-2+\\sin\\left(θ\\right)+2\\left(1-\\sin^{2}\\left(θ\\right)\\right):{\\quad}\\sin\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)$$", "input": "-2+\\sin\\left(θ\\right)+2\\left(1-\\sin^{2}\\left(θ\\right)\\right)", "result": "=\\sin\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)", "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": "=-2+\\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\\:1-2\\sin^{2}\\left(θ\\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(θ\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hukZBvFwZs1yT1SZ13MdEX4SjFdw4ywoSr8zCm/BED7MwViaLUXkeD+JukROhWdjgcntOmDfutprnf20tvBPT+5byrQDQVCXUD0vH/fvOdz8bYA0b6V2RSTOZ7Os9NODUjzITTnuulzcjZNU46Vfk0eaaWaDKlsU/T4ySFi89C4=" } }, { "type": "interim", "title": "Simplify $$-2+\\sin\\left(θ\\right)+2-2\\sin^{2}\\left(θ\\right):{\\quad}\\sin\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)$$", "input": "-2+\\sin\\left(θ\\right)+2-2\\sin^{2}\\left(θ\\right)", "result": "=\\sin\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=\\sin\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)-2+2" }, { "type": "step", "primary": "$$-2+2=0$$", "result": "=\\sin\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+i/8bRLspuub9b2O3XmF61n7z5qdIrV7mJM76che847TLx8mOdHYVzxX643JqKFIQslTDKxOR/6J+ZOGvUcautUvDL4hJNoEMxhM4xQAK/cL8HRwWibp9PO5CWWe9BltHjb2+5NLFZrsH9fcPWg/Tb7eG2mY6TTdMjTI4csedSgQO/U7SBir5BFu2QLrf0B8vRWS30fgNgDV0ICqvsbNCQ==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sPnTRO2kezWAJDlnDkg3PSOfCqr7IUJLpLErMrWAEjsR7UudhE/vdPBB2G18LKb/gGFMQkUTMNnEnFExdO5KvDLzJz4rkYnjv1/W3glakSm1s1RQOlcS2PC6wMP3eHirdsnNB6jyeqqZK42cEEzujLBaeLmMXjRC2A2WsLuccF/vbBmbuQNTF0TphKZ8RuvaBqj/v1IHPU7zVkdyHxWbhPZrYfsaGAOMmqBp2vfNYKaGfsvmBMFTMXiD5T+wtHpK" } }, { "type": "interim", "title": "Solve by substitution", "input": "\\sin\\left(θ\\right)-2\\sin^{2}\\left(θ\\right)=0", "result": "\\sin\\left(θ\\right)=0,\\:\\sin\\left(θ\\right)=\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(θ\\right)=u$$", "result": "u-2u^{2}=0" }, { "type": "interim", "title": "$$u-2u^{2}=0{\\quad:\\quad}u=0,\\:u=\\frac{1}{2}$$", "input": "u-2u^{2}=0", "steps": [ { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "-2u^{2}+u=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "-2u^{2}+u=0", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}}{2\\left(-2\\right)}", "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 $${\\quad}a=-2,\\:b=1,\\:c=0$$", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}}{2\\left(-2\\right)}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}=1$$", "input": "\\sqrt{1^{2}-4\\left(-2\\right)\\cdot\\:0}", "result": "{u}_{1,\\:2}=\\frac{-1\\pm\\:1}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "secondary": [ "$$1^{2}=1$$" ], "result": "=\\sqrt{1-4\\left(-2\\right)\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{1+4\\cdot\\:2\\cdot\\:0}" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=\\sqrt{1+0}" }, { "type": "step", "primary": "Add the numbers: $$1+0=1$$", "result": "=\\sqrt{1}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Uac2nmABaurcWf8xH8OQyyBqAK+u1uUA6eWfx20RApzehkKrn0era9rz8TlL+x/vj+lG4r/2yK6aJdxvUCCu4btCR5dIjxQ5ASg+ZPFVSsdlyvzZvmgr6BbflRzL8H8tZyb36vpKwrPyssOZsvwtrg==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-1+1}{2\\left(-2\\right)},\\:{u}_{2}=\\frac{-1-1}{2\\left(-2\\right)}" }, { "type": "interim", "title": "$$u=\\frac{-1+1}{2\\left(-2\\right)}:{\\quad}0$$", "input": "\\frac{-1+1}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1+1}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-1+1=0$$", "result": "=\\frac{0}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{0}{-4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{0}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=-0" }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZILxh0n9XGu+UX30rV52KVBraIXtDlgD3G/CwhQUjohwkKGJWEPFPk38sdJMsyPIc1L1JfkzeAMH8Sv8wAfVX6QS8+Ejzws6A1XwOMup5uLxfayEPhINvNr8uCW/1LTC" } }, { "type": "interim", "title": "$$u=\\frac{-1-1}{2\\left(-2\\right)}:{\\quad}\\frac{1}{2}$$", "input": "\\frac{-1-1}{2\\left(-2\\right)}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\frac{-1-1}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Subtract the numbers: $$-1-1=-2$$", "result": "=\\frac{-2}{-2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{-2}{-4}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CoFrCplKlG9JQtC7YmAoI1BraIXtDlgD3G/CwhQUjohwkKGJWEPFPk38sdJMsyPI4zPT6LZm4vvilNzqSUf5kUZjT4S6OaNQYcHY9HJJkC4fxUj6O9O80LGkpcT/GM7iwhKbDnZDRLLlBw2jEV2ywg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=0,\\:u=\\frac{1}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(θ\\right)$$", "result": "\\sin\\left(θ\\right)=0,\\:\\sin\\left(θ\\right)=\\frac{1}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(θ\\right)=0{\\quad:\\quad}θ=2πn,\\:θ=π+2πn$$", "input": "\\sin\\left(θ\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(θ\\right)=0$$", "result": "θ=0+2πn,\\:θ=π+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=0+2πn,\\:θ=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$θ=0+2πn:{\\quad}θ=2πn$$", "input": "θ=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "θ=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=2πn,\\:θ=π+2πn" } ], "meta": { "interimType": "N/A" } }, { "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": "step", "primary": "Combine all the solutions", "result": "θ=2πn,\\:θ=π+2πn,\\:θ=\\frac{π}{6}+2πn,\\:θ=\\frac{5π}{6}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "2\\cos^{2}(θ)+\\sin(θ)-2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }