0=-6csc(x)cot(x)

{ "query": { "display": "$$0=-6\\csc\\left(x\\right)\\cot\\left(x\\right)$$", "symbolab_question": "EQUATION#0=-6\\csc(x)\\cot(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+πn", "degrees": "x=90^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$0=-6\\csc\\left(x\\right)\\cot\\left(x\\right){\\quad:\\quad}x=\\frac{π}{2}+πn$$", "input": "0=-6\\csc\\left(x\\right)\\cot\\left(x\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "-6\\csc\\left(x\\right)\\cot\\left(x\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\csc\\left(x\\right)=0\\lor\\:\\cot\\left(x\\right)=0" }, { "type": "interim", "title": "$$\\csc\\left(x\\right)=0{\\quad:\\quad}$$No Solution", "input": "\\csc\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "$$\\csc\\left(x\\right)\\le-1\\lor\\csc\\left(x\\right)\\ge1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\cot\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+πn$$", "input": "\\cot\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cot\\left(x\\right)=0$$", "result": "x=\\frac{π}{2}+πn", "steps": [ { "type": "step", "primary": "$$\\cot\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\cot(x)\\\\\\hline 0&\\mp\\infty\\\\\\hline \\frac{π}{6}&\\sqrt{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\sqrt{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{2}+πn" } ], "meta": { "interimType": "Trig General Solutions cot 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\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": "x", "plotRequest": "0+6\\csc(x)\\cot(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }