3cos^2(x/2)=3

{ "query": { "display": "$$3\\cos^{2}\\left(\\frac{x}{2}\\right)=3$$", "symbolab_question": "EQUATION#3\\cos^{2}(\\frac{x}{2})=3" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=4πn,x=2π+4πn", "degrees": "x=0^{\\circ }+720^{\\circ }n,x=360^{\\circ }+720^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$3\\cos^{2}\\left(\\frac{x}{2}\\right)=3{\\quad:\\quad}x=4πn,\\:x=2π+4πn$$", "input": "3\\cos^{2}\\left(\\frac{x}{2}\\right)=3", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "3\\cos^{2}\\left(\\frac{x}{2}\\right)=3", "result": "\\cos\\left(\\frac{x}{2}\\right)=1,\\:\\cos\\left(\\frac{x}{2}\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(\\frac{x}{2}\\right)=u$$", "result": "3u^{2}=3" }, { "type": "interim", "title": "$$3u^{2}=3{\\quad:\\quad}u=1,\\:u=-1$$", "input": "3u^{2}=3", "steps": [ { "type": "interim", "title": "Divide both sides by $$3$$", "input": "3u^{2}=3", "result": "u^{2}=1", "steps": [ { "type": "step", "primary": "Divide both sides by $$3$$", "result": "\\frac{3u^{2}}{3}=\\frac{3}{3}" }, { "type": "step", "primary": "Simplify", "result": "u^{2}=1" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bPMyneU6LXHviR2M5I4mDFMSY3sTrae4O9mPW8Ac73M8BM0yvajEwZUfPEKsYNjoDojsf8lKRTEudrXmZFtoNk59rDIDPN4aS8VaxiqiDF9bwVz5T7/Dyunz8G+H2e0kUsD+pIkOCIZyIU4q8atffPpdu80I8/0M2UIpmdiK+fwUz/X6eKMNBZYRBdo98SOA7IZNDZV4af23xl/EbOWYixFiT9Ff2vQP/cX4TGUmxkExIVz+FKBiI0adhLuYA2iyh0QcTTXseCoUAvamq+ZQTl69HmqA496b8cWJlDcbSyUMCgLu2d7Tf0YCz45c8kcPIuTqDTaa6kFdlIEFBhEqhZb44akLOVySZB79hkFBOoJXKwDrETn4lYD/sLVsodGn7Yb8ChKEd5LBzR2wpK2+pHMAyH8W6/z6tr1o7FwXaZj/hnO7jQ5+axTIg8r+vZ4igvFWX0cLySOsKykwE1F/HVPXMMFcLeA6K7wasqiMrLZLAoAUCz4i2a7JQwVAmK4v" } }, { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{1},\\:u=-\\sqrt{1}" }, { "type": "interim", "title": "$$\\sqrt{1}=1$$", "input": "\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KfzlHGGU7KN8vfEO0eL8NN13jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fScubCnYZOJ5L8/2gsdymw1PSOscTE6qsKVI9GkIdY/eI=" } }, { "type": "interim", "title": "$$-\\sqrt{1}=-1$$", "input": "-\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kWDE1Jsjy5jGSP2mctwcnCAn9lkDfZkicUGkO3EF+IpIQKToZa7Vmz9RWrIHzooCMHIu6EZfZrJ7HpyNTqg74lPlyk515FWfACaTxs0eUEM=" } }, { "type": "step", "result": "u=1,\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(\\frac{x}{2}\\right)$$", "result": "\\cos\\left(\\frac{x}{2}\\right)=1,\\:\\cos\\left(\\frac{x}{2}\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(\\frac{x}{2}\\right)=1{\\quad:\\quad}x=4πn$$", "input": "\\cos\\left(\\frac{x}{2}\\right)=1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(\\frac{x}{2}\\right)=1$$", "result": "\\frac{x}{2}=0+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "\\frac{x}{2}=0+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=0+2πn:{\\quad}x=4πn$$", "input": "\\frac{x}{2}=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "\\frac{x}{2}=2πn" }, { "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": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=4πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\cos\\left(\\frac{x}{2}\\right)=-1{\\quad:\\quad}x=2π+4πn$$", "input": "\\cos\\left(\\frac{x}{2}\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(\\frac{x}{2}\\right)=-1$$", "result": "\\frac{x}{2}=π+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "\\frac{x}{2}=π+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "interim", "title": "Solve $$\\frac{x}{2}=π+2πn:{\\quad}x=2π+4πn$$", "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": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2π+4πn" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=4πn,\\:x=2π+4πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "3\\cos^{2}(\\frac{x}{2})-3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }