cos(1110)

{ "query": { "display": "$$\\cos\\left(1110^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\cos(1110^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{\\sqrt{3}}{2}", "decimal": "0.86602…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(1110^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(1110^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "$$\\cos\\left(1110^{\\circ\\:}\\right)=\\cos\\left(30^{\\circ\\:}\\right)$$", "input": "\\cos\\left(1110^{\\circ\\:}\\right)", "result": "=\\cos\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$1110^{\\circ\\:}$$ as $$360^{\\circ\\:}\\cdot\\:3+30^{\\circ\\:}$$", "result": "=\\cos\\left(360^{\\circ\\:}3+30^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\cos$$: $$\\cos\\left(x+360^{\\circ\\:}\\cdot\\:k\\right)=\\cos\\left(x\\right)$$", "secondary": [ "$$\\cos\\left(360^{\\circ\\:}\\cdot\\:3+30^{\\circ\\:}\\right)=\\cos\\left(30^{\\circ\\:}\\right)$$" ], "result": "=\\cos\\left(30^{\\circ\\:}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(30^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&180^{\\circ }&-1\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{2}&210^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{1}{2}&240^{\\circ }&-\\frac{1}{2}\\\\\\hline 90^{\\circ }&0&270^{\\circ }&0\\\\\\hline 120^{\\circ }&-\\frac{1}{2}&300^{\\circ }&\\frac{1}{2}\\\\\\hline 135^{\\circ }&-\\frac{\\sqrt{2}}{2}&315^{\\circ }&\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{2}&330^{\\circ }&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }