sec(495)

{ "query": { "display": "$$\\sec\\left(495^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sec(495^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-\\sqrt{2}", "decimal": "-1.41421…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sec\\left(495^{\\circ\\:}\\right)=-\\sqrt{2}$$", "input": "\\sec\\left(495^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "$$\\sec\\left(495^{\\circ\\:}\\right)=\\sec\\left(135^{\\circ\\:}\\right)$$", "input": "\\sec\\left(495^{\\circ\\:}\\right)", "result": "=\\sec\\left(135^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$495^{\\circ\\:}$$ as $$360^{\\circ\\:}+135^{\\circ\\:}$$", "result": "=\\sec\\left(360^{\\circ\\:}+135^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\sec$$: $$\\sec\\left(x+360^{\\circ\\:}\\right)=\\sec\\left(x\\right)$$", "secondary": [ "$$\\sec\\left(360^{\\circ\\:}+135^{\\circ\\:}\\right)=\\sec\\left(135^{\\circ\\:}\\right)$$" ], "result": "=\\sec\\left(135^{\\circ\\:}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sec\\left(135^{\\circ\\:}\\right)=-\\sqrt{2}$$", "input": "\\sec\\left(135^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\sec\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sec(x)&x&\\sec(x)\\\\\\hline 0&1&180^{\\circ }&-1\\\\\\hline 30^{\\circ }&\\frac{2\\sqrt{3}}{3}&210^{\\circ }&-\\frac{2\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&\\sqrt{2}&225^{\\circ }&-\\sqrt{2}\\\\\\hline 60^{\\circ }&2&240^{\\circ }&-2\\\\\\hline 90^{\\circ }&\\mathrm{Undefined}&270^{\\circ }&\\mathrm{Undefined}\\\\\\hline 120^{\\circ }&-2&300^{\\circ }&2\\\\\\hline 135^{\\circ }&-\\sqrt{2}&315^{\\circ }&\\sqrt{2}\\\\\\hline 150^{\\circ }&-\\frac{2\\sqrt{3}}{3}&330^{\\circ }&\\frac{2\\sqrt{3}}{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=-\\sqrt{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=-\\sqrt{2}" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }