2sin(x/2)=sqrt(2)

{ "query": { "display": "$$2\\sin\\left(\\frac{x}{2}\\right)=\\sqrt{2}$$", "symbolab_question": "EQUATION#2\\sin(\\frac{x}{2})=\\sqrt{2}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+4πn,x=\\frac{3π}{2}+4πn", "degrees": "x=90^{\\circ }+720^{\\circ }n,x=270^{\\circ }+720^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\sin\\left(\\frac{x}{2}\\right)=\\sqrt{2}{\\quad:\\quad}x=\\frac{π}{2}+4πn,\\:x=\\frac{3π}{2}+4πn$$", "input": "2\\sin\\left(\\frac{x}{2}\\right)=\\sqrt{2}", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\sin\\left(\\frac{x}{2}\\right)=\\sqrt{2}", "result": "\\sin\\left(\\frac{x}{2}\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\sin\\left(\\frac{x}{2}\\right)}{2}=\\frac{\\sqrt{2}}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(\\frac{x}{2}\\right)=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(\\frac{x}{2}\\right)=\\frac{\\sqrt{2}}{2}$$", "result": "\\frac{x}{2}=\\frac{π}{4}+2πn,\\:\\frac{x}{2}=\\frac{3π}{4}+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": "\\frac{x}{2}=\\frac{π}{4}+2πn,\\:\\frac{x}{2}=\\frac{3π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=\\frac{π}{4}+2πn:{\\quad}x=\\frac{π}{2}+4πn$$", "input": "\\frac{x}{2}=\\frac{π}{4}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=\\frac{π}{4}+2πn", "result": "x=\\frac{π}{2}+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{4}+2\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=2\\cdot\\:\\frac{π}{4}+2\\cdot\\:2πn", "result": "x=\\frac{π}{2}+4πn", "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 $$2\\cdot\\:\\frac{π}{4}+2\\cdot\\:2πn:{\\quad}\\frac{π}{2}+4πn$$", "input": "2\\cdot\\:\\frac{π}{4}+2\\cdot\\: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": "interim", "title": "$$2\\cdot\\:2πn=4πn$$", "input": "2\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73+wEiPbaLObTqbUS2D3M3FXTSum/z5kLpMzXS1UJIewNLJBFHVNJXcF3Tu8Q3iPc3sgi2dqxE8XWitN20kuh9g/+xmM08uF4rRxptcnmtxywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{π}{2}+4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvinBApXLcWL+El7XiR8FT7Uz+n/+FjN833TADO2gxs6I7zMFYmi1F5Hg/ibpEToVnY19WUvKsO7rj7RhU2Es3K4dPl2Rdy6SZiOVUpWNZcMJC7kAjP76qW66lOUsURwT0nU2tJAL4GDvnfdGiGxVTy+W+i7ODJGGicZ+Q8385Fpbge/EW6scPGHMpRVZCtwbXWQ==" } }, { "type": "step", "result": "x=\\frac{π}{2}+4πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=\\frac{3π}{4}+2πn:{\\quad}x=\\frac{3π}{2}+4πn$$", "input": "\\frac{x}{2}=\\frac{3π}{4}+2πn", "steps": [ { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{x}{2}=\\frac{3π}{4}+2πn", "result": "x=\\frac{3π}{2}+4πn", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2x}{2}=2\\cdot\\:\\frac{3π}{4}+2\\cdot\\:2πn" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=2\\cdot\\:\\frac{3π}{4}+2\\cdot\\:2πn", "result": "x=\\frac{3π}{2}+4πn", "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 $$2\\cdot\\:\\frac{3π}{4}+2\\cdot\\:2πn:{\\quad}\\frac{3π}{2}+4πn$$", "input": "2\\cdot\\:\\frac{3π}{4}+2\\cdot\\:2πn", "steps": [ { "type": "interim", "title": "$$2\\cdot\\:\\frac{3π}{4}=\\frac{3π}{2}$$", "input": "2\\cdot\\:\\frac{3π}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3π2}{4}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=\\frac{6π}{4}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{3π}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvipJkMcY2bVMLBmUTz4hhEZR1g99dC9fj9sg0EHzBIRDRN7VgspnUW4RekHLf5B6EmF5NkzKQgtswLlLi9MgL+gruVdKnRtF96VRP4iLsEi1Yxn9B12H513zzgATnStkPPONv3CS+rhjiRitEVTwxNoc=" } }, { "type": "interim", "title": "$$2\\cdot\\:2πn=4πn$$", "input": "2\\cdot\\:2πn", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73+wEiPbaLObTqbUS2D3M3FXTSum/z5kLpMzXS1UJIewNLJBFHVNJXcF3Tu8Q3iPc3sgi2dqxE8XWitN20kuh9g/+xmM08uF4rRxptcnmtxywiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{3π}{2}+4πn" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qviijMwzUbCAxl7oYZw39vLPPj+O38RAvKhZolrKec34mYcJChiVhDxT5N/LHSTLMjyAgSGGqcEVHvfOvxKWupqfZwkTBskMok+QnwLC7by1E9ixpjZyz1hn/Tx95aZo6CSBaMN9uKsa/JPEDniyiviLdwwhymJyPZ2eGwNJAfCbGPadl0eCLqgPgD3O2XqRYdVQ==" } }, { "type": "step", "result": "x=\\frac{3π}{2}+4πn" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{π}{2}+4πn,\\:x=\\frac{3π}{2}+4πn" } ], "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(\\frac{x}{2})-\\sqrt{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }