sin^2(x)=3sin(x)-2

{ "query": { "display": "$$\\sin^{2}\\left(x\\right)=3\\sin\\left(x\\right)-2$$", "symbolab_question": "EQUATION#\\sin^{2}(x)=3\\sin(x)-2" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+2πn", "degrees": "x=90^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{2}\\left(x\\right)=3\\sin\\left(x\\right)-2{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin^{2}\\left(x\\right)=3\\sin\\left(x\\right)-2", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{2}\\left(x\\right)=3\\sin\\left(x\\right)-2", "result": "\\sin\\left(x\\right)=2,\\:\\sin\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{2}=3u-2" }, { "type": "interim", "title": "$$u^{2}=3u-2{\\quad:\\quad}u=2,\\:u=1$$", "input": "u^{2}=3u-2", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the left side", "input": "u^{2}=3u-2", "result": "u^{2}+2=3u", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "u^{2}+2=3u-2+2" }, { "type": "step", "primary": "Simplify", "result": "u^{2}+2=3u" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$3u\\:$$to the left side", "input": "u^{2}+2=3u", "result": "u^{2}+2-3u=0", "steps": [ { "type": "step", "primary": "Subtract $$3u$$ from both sides", "result": "u^{2}+2-3u=3u-3u" }, { "type": "step", "primary": "Simplify", "result": "u^{2}+2-3u=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kc+1o2FCpNacOShlarQvXALz44h/S+6E6JYQRqyafhaEe0atupXBi79RbtbDh8ipMxCfkUix29Opj0y0EBnvASYczQvYXcZ2t/UTMwYL1+5X/0VCBpEeIxKQa2H0p0Wzic2+FidoCLUBtPd5P/cqkJ9AzlAioTbPkbJdA2+8wLotK7Ea7uQhfSgir3oXD6HA+tSH6ZEUVRk7XVBvI/9NAvgotvVuraanirIs6PwFjcVwanLR4+0ctcR7z+Qn2FhgbRc/6QB7RaMJy83FmfHiysQVTIXxHbnuqrGIe5Uv+ELDIUjprCA+XIptxfb/EbOOB4CES9nGxD2S5ut+HDkiOxtWKyJwU+3om7eHGIbx8A80UCRHZyRKDzQfKBxnNPRTvJXrhRQTKZrV0QqJ1oeAGfve1FkLGYy7pfNHW7ge8PWtSON6otHROlKM8B29ZaSGfY/p/rzlH/IGeSreLVwnL4LI9sZgZtGXglO6wn2kzD+/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "u^{2}-3u+2=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "u^{2}-3u+2=0", "result": "{u}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}}{2\\cdot\\:1}", "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 $${\\quad}a=1,\\:b=-3,\\:c=2$$", "result": "{u}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}=1$$", "input": "\\sqrt{\\left(-3\\right)^{2}-4\\cdot\\:1\\cdot\\:2}", "result": "{u}_{1,\\:2}=\\frac{-\\left(-3\\right)\\pm\\:1}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-3\\right)^{2}=3^{2}$$" ], "result": "=\\sqrt{3^{2}-4\\cdot\\:1\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:2=8$$", "result": "=\\sqrt{3^{2}-8}" }, { "type": "step", "primary": "$$3^{2}=9$$", "result": "=\\sqrt{9-8}" }, { "type": "step", "primary": "Subtract the numbers: $$9-8=1$$", "result": "=\\sqrt{1}" }, { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s77dtxLd9UeyppCvogsL4v83GW/jTwUrqHAcqr1cHDsBBw4ba/3KCEGXsM04IlGc/HzMFYmi1F5Hg/ibpEToVnY247KVn/9CN730bZvRQSpulUlJGhz/9/49m9VyFXtj5ecZb+NPBSuocByqvVwcOwEEDVEq/Lyx8C1I/8Q6ctddo=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\left(-3\\right)+1}{2\\cdot\\:1},\\:{u}_{2}=\\frac{-\\left(-3\\right)-1}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$u=\\frac{-\\left(-3\\right)+1}{2\\cdot\\:1}:{\\quad}2$$", "input": "\\frac{-\\left(-3\\right)+1}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{3+1}{2\\cdot\\:1}" }, { "type": "step", "primary": "Add the numbers: $$3+1=4$$", "result": "=\\frac{4}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7BRRger6gkjQuZFNUJO1CQrJqVPiSqeELumwKNYa9UokgJ/ZZA32ZInFBpDtxBfiKRcASOqRpLIeIyUBzgC+nUCr6EdYdh/n2c4DPMiuUGOKGT1o1ywYsZORt5Dm0dqjD1FYlXu7xkLLsgM93YHqktw==" } }, { "type": "interim", "title": "$$u=\\frac{-\\left(-3\\right)-1}{2\\cdot\\:1}:{\\quad}1$$", "input": "\\frac{-\\left(-3\\right)-1}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{3-1}{2\\cdot\\:1}" }, { "type": "step", "primary": "Subtract the numbers: $$3-1=2$$", "result": "=\\frac{2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fb8/QwpiZ9fKB0o6NPD4MrJqVPiSqeELumwKNYa9UokgJ/ZZA32ZInFBpDtxBfiK7J5E5gGi2xwchkRMjoVJ7ir6EdYdh/n2c4DPMiuUGOJoUL/JiggAW6L5OHgO2bRY1FYlXu7xkLLsgM93YHqktw==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=2,\\:u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=2,\\:\\sin\\left(x\\right)=1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=2{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=2", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=1{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=1$$", "result": "x=\\frac{π}{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": "x=\\frac{π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{2}+2πn" } ], "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)-3\\sin(x)+2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }