2sin^2(1/2 x)+C=sin(x)+C

{ "query": { "display": "$$2\\sin^{2}\\left(\\frac{1}{2}x\\right)+C=\\sin\\left(x\\right)+C$$", "symbolab_question": "EQUATION#2\\sin^{2}(\\frac{1}{2}x)+C=\\sin(x)+C" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=4πn,x=2π+4πn,x=\\frac{π}{2}+2πn", "degrees": "x=0^{\\circ }+720^{\\circ }n,x=360^{\\circ }+720^{\\circ }n,x=90^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\sin^{2}\\left(\\frac{1}{2}x\\right)+C=\\sin\\left(x\\right)+C{\\quad:\\quad}x=4πn,\\:x=2π+4πn,\\:x=\\frac{π}{2}+2πn$$", "input": "2\\sin^{2}\\left(\\frac{1}{2}x\\right)+C=\\sin\\left(x\\right)+C", "steps": [ { "type": "step", "primary": "Subtract $$\\sin\\left(x\\right)+C$$ from both sides", "result": "2\\sin^{2}\\left(\\frac{x}{2}\\right)-\\sin\\left(x\\right)=0" }, { "type": "step", "primary": "Let: $$u=\\frac{x}{2}$$", "result": "2\\sin^{2}\\left(u\\right)-\\sin\\left(2u\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "-\\sin\\left(2u\\right)+2\\sin^{2}\\left(u\\right)", "result": "2\\sin^{2}\\left(u\\right)-2\\cos\\left(u\\right)\\sin\\left(u\\right)=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=-2\\sin\\left(u\\right)\\cos\\left(u\\right)+2\\sin^{2}\\left(u\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7SHDS6kJwJIV4q/3HJvK1htWXrwzJU3/10rsxY60+peIpzzLHu1HsPMRxRdIZX4sFPO0XSKqaWlOK6mHDvVM10YIYWgyd6SbL0eRRsTHoy/YnSKF5/4+51qVY0U4KnLmx3pacYxllh6Ga1xVWDSxTWWDjYlIGM0C2R2oMCFCGHxGCZ7NeGaBBoyvJvw0OqO0OiD9kx/VS5qIswUMHVF3GBDMJZkL/6j5jtVUOIJSBOKqlxYnWxJTu1x/Pjz4hDH7Q" } }, { "type": "interim", "title": "Factor $$2\\sin^{2}\\left(u\\right)-2\\cos\\left(u\\right)\\sin\\left(u\\right):{\\quad}2\\sin\\left(u\\right)\\left(\\sin\\left(u\\right)-\\cos\\left(u\\right)\\right)$$", "input": "2\\sin^{2}\\left(u\\right)-2\\cos\\left(u\\right)\\sin\\left(u\\right)", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$\\sin^{2}\\left(u\\right)=\\sin\\left(u\\right)\\sin\\left(u\\right)$$" ], "result": "=2\\sin\\left(u\\right)\\sin\\left(u\\right)-2\\sin\\left(u\\right)\\cos\\left(u\\right)", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$2\\sin\\left(u\\right)$$", "result": "=2\\sin\\left(u\\right)\\left(\\sin\\left(u\\right)-\\cos\\left(u\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "2\\sin\\left(u\\right)\\left(\\sin\\left(u\\right)-\\cos\\left(u\\right)\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(u\\right)=0\\lor\\:\\sin\\left(u\\right)-\\cos\\left(u\\right)=0" }, { "type": "interim", "title": "$$\\sin\\left(u\\right)=0{\\quad:\\quad}u=2πn,\\:u=π+2πn$$", "input": "\\sin\\left(u\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(u\\right)=0$$", "result": "u=0+2πn,\\:u=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": "u=0+2πn,\\:u=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$u=0+2πn:{\\quad}u=2πn$$", "input": "u=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "u=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "u=2πn,\\:u=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sin\\left(u\\right)-\\cos\\left(u\\right)=0{\\quad:\\quad}u=\\frac{π}{4}+πn$$", "input": "\\sin\\left(u\\right)-\\cos\\left(u\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(u\\right)-\\cos\\left(u\\right)=0", "result": "\\tan\\left(u\\right)-1=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\cos\\left(u\\right),\\:\\cos\\left(u\\right)\\neq0$$", "result": "\\frac{\\sin\\left(u\\right)-\\cos\\left(u\\right)}{\\cos\\left(u\\right)}=\\frac{0}{\\cos\\left(u\\right)}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{\\sin\\left(u\\right)}{\\cos\\left(u\\right)}-1=0" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\tan\\left(x\\right)$$", "result": "\\tan\\left(u\\right)-1=0" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFEggbwrWuqWdEQExoS7+Ui8FMS0G0HgjUXwc3RQOmn9BYznLdIBaHD4ibBnEILJTDXUHW1mE5UQUTIwuQd6ag+zflewpXsN2Z5/tCbT2a61d9BnHcjXjY3HWkM0Psuc55E3kCh3oevUunZ7/b0qFKBTCPNlUYj52tc4Y8B80I30B5l60FyrIpKpS8qSplgYxPCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\tan\\left(u\\right)-1=0", "result": "\\tan\\left(u\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\tan\\left(u\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "\\tan\\left(u\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\tan\\left(u\\right)=1$$", "result": "u=\\frac{π}{4}+πn", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "u=\\frac{π}{4}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "u=2πn,\\:u=π+2πn,\\:u=\\frac{π}{4}+πn" }, { "type": "step", "primary": "Substitute back $$u=\\frac{x}{2}$$" }, { "type": "interim", "title": "$$\\frac{x}{2}=2πn{\\quad:\\quad}x=4πn$$", "input": "\\frac{x}{2}=2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=2πn", "result": "x=4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "x=4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$\\frac{x}{2}=π+2πn{\\quad:\\quad}x=2π+4πn$$", "input": "\\frac{x}{2}=π+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=π+2πn", "result": "x=2π+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2π+2\\cdot\\:2πn" }, { "type": "step", "primary": "Simplify", "result": "x=2π+4πn" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "interim", "title": "$$\\frac{x}{2}=\\frac{π}{4}+πn{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\frac{x}{2}=\\frac{π}{4}+πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=\\frac{π}{4}+πn", "result": "x=\\frac{π}{2}+2πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{4}+2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{4}+2πn", "result": "x=\\frac{π}{2}+2πn", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{π}{4}+2πn:{\\quad}\\frac{π}{2}+2πn$$", "input": "2\\cdot\\:\\frac{π}{4}+2πn", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{π}{4}=\\frac{π}{2}$$", "input": "2\\cdot\\:\\frac{π}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{π2}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvippetnwDw0F99AVSOZ/vPAwJQJZuTAY5js+oqjdT8kslZguKS1b2cgc0s2D0HaMML/8//6/nV5O4fb8Xgwi7maq51BdaKQokHx8SZGXrDp2iTpbCyD7Gyql0g1vQY+8LMm6sJzS/r5J7Nekm75J11/g=" } }, { "type": "step", "result": "=\\frac{π}{2}+2πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvip1vBwJwamFqy+lWvv87QacgJ/ZZA32ZInFBpDtxBfiKXYGCmiBF99lesmXZ9iIfJ7VOu/yv+MQkxgCBvT6c/TmBBTEk/JQ2cZ9WKuRzClU7hdlJPcZzzyhBqDm8dNgG19MzzULovGGcpsEyVazFzb/WqV0Mz8uq7u3M19r+Y70F" } }, { "type": "step", "result": "x=\\frac{π}{2}+2πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "result": "x=4πn,\\:x=2π+4πn,\\: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": "2\\sin^{2}(\\frac{1}{2}x)+C=\\sin(x)+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }