4cos(θ)=sqrt(2)+2cos(θ)

{ "query": { "display": "$$4\\cos\\left(θ\\right)=\\sqrt{2}+2\\cos\\left(θ\\right)$$", "symbolab_question": "EQUATION#4\\cos(θ)=\\sqrt{2}+2\\cos(θ)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{π}{4}+2πn,θ=\\frac{7π}{4}+2πn", "degrees": "θ=45^{\\circ }+360^{\\circ }n,θ=315^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$4\\cos\\left(θ\\right)=\\sqrt{2}+2\\cos\\left(θ\\right){\\quad:\\quad}θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn$$", "input": "4\\cos\\left(θ\\right)=\\sqrt{2}+2\\cos\\left(θ\\right)", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "4\\cos\\left(θ\\right)=\\sqrt{2}+2\\cos\\left(θ\\right)", "result": "\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(θ\\right)=u$$", "result": "4u=\\sqrt{2}+2u" }, { "type": "interim", "title": "$$4u=\\sqrt{2}+2u{\\quad:\\quad}u=\\frac{\\sqrt{2}}{2}$$", "input": "4u=\\sqrt{2}+2u", "steps": [ { "type": "interim", "title": "Move $$2u\\:$$to the left side", "input": "4u=\\sqrt{2}+2u", "result": "2u=\\sqrt{2}", "steps": [ { "type": "step", "primary": "Subtract $$2u$$ from both sides", "result": "4u-2u=\\sqrt{2}+2u-2u" }, { "type": "step", "primary": "Simplify", "result": "2u=\\sqrt{2}" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2u=\\sqrt{2}", "result": "u=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2u}{2}=\\frac{\\sqrt{2}}{2}" }, { "type": "step", "primary": "Simplify", "result": "u=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(θ\\right)$$", "result": "\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}{\\quad:\\quad}θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn$$", "input": "\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(θ\\right)=\\frac{\\sqrt{2}}{2}$$", "result": "θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+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": "θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=\\frac{π}{4}+2πn,\\:θ=\\frac{7π}{4}+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "4\\cos(θ)-\\sqrt{2}-2\\cos(θ)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }