csc(750)

{ "query": { "display": "$$\\csc\\left(750^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\csc(750^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "2", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\csc\\left(750^{\\circ\\:}\\right)=2$$", "input": "\\csc\\left(750^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "$$\\csc\\left(750^{\\circ\\:}\\right)=\\csc\\left(30^{\\circ\\:}\\right)$$", "input": "\\csc\\left(750^{\\circ\\:}\\right)", "result": "=\\csc\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$750^{\\circ\\:}$$ as $$360^{\\circ\\:}\\cdot\\:2+30^{\\circ\\:}$$", "result": "=\\csc\\left(360^{\\circ\\:}2+30^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\csc$$: $$\\csc\\left(x+360^{\\circ\\:}\\cdot\\:k\\right)=\\csc\\left(x\\right)$$", "secondary": [ "$$\\csc\\left(360^{\\circ\\:}\\cdot\\:2+30^{\\circ\\:}\\right)=\\csc\\left(30^{\\circ\\:}\\right)$$" ], "result": "=\\csc\\left(30^{\\circ\\:}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\csc\\left(30^{\\circ\\:}\\right)=2$$", "input": "\\csc\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\csc\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\csc(x)&x&\\csc(x)\\\\\\hline 0&\\mathrm{Undefiend}&180^{\\circ }&\\mathrm{Undefiend}\\\\\\hline 30^{\\circ }&2&210^{\\circ }&-2\\\\\\hline 45^{\\circ }&\\sqrt{2}&225^{\\circ }&-\\sqrt{2}\\\\\\hline 60^{\\circ }&\\frac{2\\sqrt{3}}{3}&240^{\\circ }&-\\frac{2\\sqrt{3}}{3}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{2\\sqrt{3}}{3}&300^{\\circ }&-\\frac{2\\sqrt{3}}{3}\\\\\\hline 135^{\\circ }&\\sqrt{2}&315^{\\circ }&-\\sqrt{2}\\\\\\hline 150^{\\circ }&2&330^{\\circ }&-2\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=2" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=2" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }