cos(3θ)=cos(θ)

{ "query": { "display": "$$\\cos\\left(3θ\\right)=\\cos\\left(θ\\right)$$", "symbolab_question": "EQUATION#\\cos(3θ)=\\cos(θ)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=πn,θ=\\frac{π}{2}+πn", "degrees": "θ=0^{\\circ }+180^{\\circ }n,θ=90^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(3θ\\right)=\\cos\\left(θ\\right){\\quad:\\quad}θ=πn,\\:θ=\\frac{π}{2}+πn$$", "input": "\\cos\\left(3θ\\right)=\\cos\\left(θ\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cos\\left(θ\\right)$$ from both sides", "result": "\\cos\\left(3θ\\right)-\\cos\\left(θ\\right)=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\cos\\left(3θ\\right)-\\cos\\left(θ\\right)", "result": "-2\\sin\\left(2θ\\right)\\sin\\left(θ\\right)=0", "steps": [ { "type": "step", "primary": "Use the Sum to Product identity: $$\\cos\\left(s\\right)-\\cos\\left(t\\right)=-2\\sin\\left(\\frac{s+t}{2}\\right)\\sin\\left(\\frac{s-t}{2}\\right)$$", "result": "=-2\\sin\\left(\\frac{3θ+θ}{2}\\right)\\sin\\left(\\frac{3θ-θ}{2}\\right)" }, { "type": "interim", "title": "Simplify $$-2\\sin\\left(\\frac{3θ+θ}{2}\\right)\\sin\\left(\\frac{3θ-θ}{2}\\right):{\\quad}-2\\sin\\left(2θ\\right)\\sin\\left(θ\\right)$$", "input": "-2\\sin\\left(\\frac{3θ+θ}{2}\\right)\\sin\\left(\\frac{3θ-θ}{2}\\right)", "result": "=-2\\sin\\left(2θ\\right)\\sin\\left(θ\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{3θ+θ}{2}=2θ$$", "input": "\\frac{3θ+θ}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$3θ+θ=4θ$$", "result": "=\\frac{4θ}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Ksg9eVsC9Fv5hGIDGXtrCdVBn2NNCFZqg4ZoVh6UwqjkVi15I8rBefLi4Iyt2wrGZHczU7dFYY134V2zeCegNLskHR1XdSSKSdbUSUwWGETerXnP+DkqGqSGcqrZnhB" } }, { "type": "step", "result": "=-2\\sin\\left(2θ\\right)\\sin\\left(\\frac{3θ-θ}{2}\\right)" }, { "type": "interim", "title": "$$\\frac{3θ-θ}{2}=θ$$", "input": "\\frac{3θ-θ}{2}", "steps": [ { "type": "step", "primary": "Add similar elements: $$3θ-θ=2θ$$", "result": "=\\frac{2θ}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78dJk74FDXkIq5rb3uAuenCdVBn2NNCFZqg4ZoVh6UwqjkVi15I8rBefLi4Iyt2wrZoPzxmrP7+4TMbrYjqofDI9Khv5Wmbz4sf9UMSx6ItQaPAx5KP2rVBoLjqshJFw7" } }, { "type": "step", "result": "=-2\\sin\\left(2θ\\right)\\sin\\left(θ\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WKQDtlNbtfOFijcUoWrkOqray957j+3C/M/hxU1pYxgPt06UMl/6THlVu8y6N9KDIGio76cBq3IcP4gORIfTNszBWJotReR4P4m6RE6FZ2P8hVcYdpEZlK2VxDJSKy5HjEcYkEcIp9OcwOnklM0L0R429vuTSxWa7B/X3D1oP03wgIWhXhtwOYmcDmqyesWQf2jblA0VOzn3rABfI+zPDvHIl82AKAiO3wI0fFiLBK4TRBs03ziTVjAdYS9MG+cr" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7lRPp6QndqgNT+L6IZutAbN5hwUnUmJybCjyUuA3KPlz1fWutlCU4zUzgkFchUVlKOeWgsE4Mk40prEDZkKQ8Xcq64+b8YguXf4qCtKW9b7ISFoL1Efx+E4fKUQsJTWquBYEidskR/oS/CmglpmWIBO7nGBp/aiM+wK6HvcGlBOx2WVG1dxuhtwcvc5+As2ToKO+d89V0ksUsI4xf7WSo4wJYYPwif7On5mZsZiNIHrw=" } }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(2θ\\right)=0\\lor\\:\\sin\\left(θ\\right)=0" }, { "type": "interim", "title": "$$\\sin\\left(2θ\\right)=0{\\quad:\\quad}θ=πn,\\:θ=\\frac{π}{2}+πn$$", "input": "\\sin\\left(2θ\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(2θ\\right)=0$$", "result": "2θ=0+2πn,\\:2θ=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": "2θ=0+2πn,\\:2θ=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$2θ=0+2πn:{\\quad}θ=πn$$", "input": "2θ=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "2θ=2πn" }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2θ=2πn", "result": "θ=πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2θ}{2}=\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "θ=πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2θ=π+2πn:{\\quad}θ=\\frac{π}{2}+πn$$", "input": "2θ=π+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2θ=π+2πn", "result": "θ=\\frac{π}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2θ}{2}=\\frac{π}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "θ=\\frac{π}{2}+πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7VmoPrHRBMRj/LKIhHmbM6cSP7F8VGulIRMZ60vJ5y+VPnie/F7TDTqTxkucfXTe/L2LifiwEM4YYN0r1csQ0NxcENkL5mTBunE57PHwSJwyITAH5gIW32oxvJMTXlnMjW8Fc+U+/w8rp8/Bvh9ntJFLA/qSJDgiGciFOKvGrX3z6XbvNCPP9DNlCKZnYivn8XUZHyFcsb+PrYH5B5FjpgTEZOhzOLFFCWOVe0TAs9PuItZ4o+IoRkWIEKrqs1tFAI/4vZTI9bEw7h05UX+rc/f0OAYY8+kziw3KxRm0ToEdpkIituavtlsGU8x87aNhZWPdNylMtyJvNuTRgR9DBgfgPuBLKtj6DifJlSlX4/TGY6NWrFE2eS1l1zZPcUejE4IMIfYNwM2JqUy6/LsGfCeT5Rj0kETm/9hJnqz4ZzPOdiqcAK81Ko4/KpyFPiCZ8dptnCLknK03/W5+ZCJKKaQYyBJL2j075R6FHVrdPUhNadgwzuY6Jb0cogSIeKVquUcbinK14Gon5f3QBcn/JKpwBSyQ/f/TDfHNs9qNDLfIpcZPZm+pU/lbBV9einQehfhvb9mimrAoYFeuVVvHE1g==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=πn,\\:θ=\\frac{π}{2}+πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "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": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=πn,\\:θ=\\frac{π}{2}+πn,\\:θ=2πn,\\:θ=π+2πn" }, { "type": "step", "primary": "Merge Overlapping Intervals", "result": "θ=πn,\\:θ=\\frac{π}{2}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\cos(3θ)-\\cos(θ)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }