sec(θ)+2=0

{ "query": { "display": "$$\\sec\\left(θ\\right)+2=0$$", "symbolab_question": "EQUATION#\\sec(θ)+2=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{2π}{3}+2πn,θ=\\frac{4π}{3}+2πn", "degrees": "θ=120^{\\circ }+360^{\\circ }n,θ=240^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sec\\left(θ\\right)+2=0{\\quad:\\quad}θ=\\frac{2π}{3}+2πn,\\:θ=\\frac{4π}{3}+2πn$$", "input": "\\sec\\left(θ\\right)+2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "\\sec\\left(θ\\right)+2=0", "result": "\\sec\\left(θ\\right)=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "\\sec\\left(θ\\right)+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "\\sec\\left(θ\\right)=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sec\\left(θ\\right)=-2$$", "result": "θ=\\frac{2π}{3}+2πn,\\:θ=\\frac{4π}{3}+2πn", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{2\\sqrt{3}}{3}&\\frac{7π}{6}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&2&\\frac{4π}{3}&-2\\\\\\hline \\frac{π}{2}&\\mathrm{Undefined}&\\frac{3π}{2}&\\mathrm{Undefined}\\\\\\hline \\frac{2π}{3}&-2&\\frac{5π}{3}&2\\\\\\hline \\frac{3π}{4}&-\\sqrt{2}&\\frac{7π}{4}&\\sqrt{2}\\\\\\hline \\frac{5π}{6}&-\\frac{2\\sqrt{3}}{3}&\\frac{11π}{6}&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "θ=\\frac{2π}{3}+2πn,\\:θ=\\frac{4π}{3}+2πn" } ], "meta": { "interimType": "Trig General Solutions sec 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\sec(θ)+2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }