cot((11pi)/2)

{ "query": { "display": "$$\\cot\\left(\\frac{11π}{2}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\cot(\\frac{11π}{2})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cot\\left(\\frac{11π}{2}\\right)=0$$", "input": "\\cot\\left(\\frac{11π}{2}\\right)", "steps": [ { "type": "interim", "title": "$$\\cot\\left(\\frac{11π}{2}\\right)=\\cot\\left(\\frac{π}{2}\\right)$$", "input": "\\cot\\left(\\frac{11π}{2}\\right)", "result": "=\\cot\\left(\\frac{π}{2}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$\\frac{11π}{2}$$ as $$π\\cdot\\:5+\\frac{π}{2}$$", "result": "=\\cot\\left(π5+\\frac{π}{2}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\cot$$: $$\\cot\\left(x+π\\cdot\\:k\\right)=\\cot\\left(x\\right)$$", "secondary": [ "$$\\cot\\left(π\\cdot\\:5+\\frac{π}{2}\\right)=\\cot\\left(\\frac{π}{2}\\right)$$" ], "result": "=\\cot\\left(\\frac{π}{2}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cot\\left(\\frac{π}{2}\\right)=0$$", "input": "\\cot\\left(\\frac{π}{2}\\right)", "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": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }