2cos(θ)+sqrt(2)=0,0<= θ<= 2pi

{ "query": { "display": "$$2\\cos\\left(θ\\right)+\\sqrt{2}=0,\\:0\\le\\:θ\\le\\:2π$$", "symbolab_question": "EQUATION#2\\cos(θ)+\\sqrt{2}=0,0\\le θ\\le 2π" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{3π}{4},θ=\\frac{5π}{4}", "degrees": "θ=135^{\\circ },θ=225^{\\circ }", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos\\left(θ\\right)+\\sqrt{2}=0,\\:0\\le\\:θ\\le\\:2π{\\quad:\\quad}θ=\\frac{3π}{4},\\:θ=\\frac{5π}{4}$$", "input": "2\\cos\\left(θ\\right)+\\sqrt{2}=0,\\:0\\le\\:θ\\le\\:2π", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{2}\\:$$to the right side", "input": "2\\cos\\left(θ\\right)+\\sqrt{2}=0", "result": "2\\cos\\left(θ\\right)=-\\sqrt{2}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{2}$$ from both sides", "result": "2\\cos\\left(θ\\right)+\\sqrt{2}-\\sqrt{2}=0-\\sqrt{2}" }, { "type": "step", "primary": "Simplify", "result": "2\\cos\\left(θ\\right)=-\\sqrt{2}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\cos\\left(θ\\right)=-\\sqrt{2}", "result": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{2}}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\cos\\left(θ\\right)}{2}=\\frac{-\\sqrt{2}}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(θ\\right)=-\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(θ\\right)=-\\frac{\\sqrt{2}}{2}$$", "result": "θ=\\frac{3π}{4}+2πn,\\:θ=\\frac{5π}{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{3π}{4}+2πn,\\:θ=\\frac{5π}{4}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:θ\\le\\:2π$$", "result": "θ=\\frac{3π}{4},\\:θ=\\frac{5π}{4}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "2\\cos(θ)+\\sqrt{2},0\\le θ\\le 2π" }, "showViewLarger": true } }, "meta": { "showVerify": true } }