arcsin(csc(-(5pi)/2))

{ "query": { "display": "$$\\arcsin\\left(\\csc\\left(-\\frac{5π}{2}\\right)\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\arcsin(\\csc(-\\frac{5π}{2}))" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-\\frac{π}{2}", "decimal": "-90", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\arcsin\\left(\\csc\\left(-\\frac{5π}{2}\\right)\\right)=-\\frac{π}{2}$$", "input": "\\arcsin\\left(\\csc\\left(-\\frac{5π}{2}\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the following property: $$\\csc\\left(-x\\right)=-\\csc\\left(x\\right)$$", "secondary": [ "$$\\csc\\left(-\\frac{5π}{2}\\right)=-\\csc\\left(\\frac{5π}{2}\\right)$$" ], "result": "=\\arcsin\\left(-\\csc\\left(\\frac{5π}{2}\\right)\\right)" }, { "type": "step", "primary": "Use the following property: $$\\arcsin\\left(-x\\right)=-\\arcsin\\left(x\\right)$$", "secondary": [ "$$\\arcsin\\left(-\\csc\\left(\\frac{5π}{2}\\right)\\right)=-\\arcsin\\left(\\csc\\left(\\frac{5π}{2}\\right)\\right)$$" ], "result": "=-\\arcsin\\left(\\csc\\left(\\frac{5π}{2}\\right)\\right)" }, { "type": "interim", "title": "$$\\csc\\left(\\frac{5π}{2}\\right)=\\csc\\left(\\frac{π}{2}\\right)$$", "input": "\\csc\\left(\\frac{5π}{2}\\right)", "result": "=-\\arcsin\\left(\\csc\\left(\\frac{π}{2}\\right)\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$\\frac{5π}{2}$$ as $$2π+\\frac{π}{2}$$", "result": "=\\csc\\left(2π+\\frac{π}{2}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\csc$$: $$\\csc\\left(x+2π\\right)=\\csc\\left(x\\right)$$", "secondary": [ "$$\\csc\\left(2π+\\frac{π}{2}\\right)=\\csc\\left(\\frac{π}{2}\\right)$$" ], "result": "=\\csc\\left(\\frac{π}{2}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\csc\\left(\\frac{π}{2}\\right)=1$$", "input": "\\csc\\left(\\frac{π}{2}\\right)", "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": "=\\arcsin\\left(1\\right)" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=-\\arcsin\\left(1\\right)" }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\arcsin\\left(1\\right)=\\frac{π}{2}$$", "input": "\\arcsin\\left(1\\right)", "steps": [ { "type": "step", "primary": "$$\\begin{array}{|c|c|c|}\\hline x&\\arcsin(x)&\\arcsin(x)\\\\\\hline 0&0&0^{\\circ}\\\\\\hline \\frac{1}{2}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline \\frac{\\sqrt{2}}{2}&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline \\frac{\\sqrt{3}}{2}&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline 1&\\frac{\\pi}{2}&90^{\\circ}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{π}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=-\\frac{π}{2}" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }