2csc(θ)+2=0

{ "query": { "display": "$$2\\csc\\left(θ\\right)+2=0$$", "symbolab_question": "EQUATION#2\\csc(θ)+2=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{3π}{2}+2πn", "degrees": "θ=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\csc\\left(θ\\right)+2=0{\\quad:\\quad}θ=\\frac{3π}{2}+2πn$$", "input": "2\\csc\\left(θ\\right)+2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "2\\csc\\left(θ\\right)+2=0", "result": "2\\csc\\left(θ\\right)=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "2\\csc\\left(θ\\right)+2-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "2\\csc\\left(θ\\right)=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2\\csc\\left(θ\\right)=-2", "result": "\\csc\\left(θ\\right)=-1", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2\\csc\\left(θ\\right)}{2}=\\frac{-2}{2}" }, { "type": "step", "primary": "Simplify", "result": "\\csc\\left(θ\\right)=-1" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\csc\\left(θ\\right)=-1$$", "result": "θ=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\csc\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\csc(x)&x&\\csc(x)\\\\\\hline 0&\\mathrm{Undefiend}&π&\\mathrm{Undefiend}\\\\\\hline \\frac{π}{6}&2&\\frac{7π}{6}&-2\\\\\\hline \\frac{π}{4}&\\sqrt{2}&\\frac{5π}{4}&-\\sqrt{2}\\\\\\hline \\frac{π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{4π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{2\\sqrt{3}}{3}&\\frac{5π}{3}&-\\frac{2\\sqrt{3}}{3}\\\\\\hline \\frac{3π}{4}&\\sqrt{2}&\\frac{7π}{4}&-\\sqrt{2}\\\\\\hline \\frac{5π}{6}&2&\\frac{11π}{6}&-2\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "θ=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions csc 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "2\\csc(θ)+2" }, "showViewLarger": true } }, "meta": { "showVerify": true } }