csc(-600)

{ "query": { "display": "$$\\csc\\left(-600^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\csc(-600^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{2\\sqrt{3}}{3}", "decimal": "1.15470…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\csc\\left(-600^{\\circ\\:}\\right)=\\frac{2\\sqrt{3}}{3}$$", "input": "\\csc\\left(-600^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Use the following property: $$\\csc\\left(-x\\right)=-\\csc\\left(x\\right)$$", "secondary": [ "$$\\csc\\left(-600^{\\circ\\:}\\right)=-\\csc\\left(600^{\\circ\\:}\\right)$$" ], "result": "=-\\csc\\left(600^{\\circ\\:}\\right)" }, { "type": "interim", "title": "$$\\csc\\left(600^{\\circ\\:}\\right)=\\csc\\left(240^{\\circ\\:}\\right)$$", "input": "\\csc\\left(600^{\\circ\\:}\\right)", "result": "=-\\csc\\left(240^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$600^{\\circ\\:}$$ as $$360^{\\circ\\:}+240^{\\circ\\:}$$", "result": "=\\csc\\left(360^{\\circ\\:}+240^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\csc$$: $$\\csc\\left(x+360^{\\circ\\:}\\right)=\\csc\\left(x\\right)$$", "secondary": [ "$$\\csc\\left(360^{\\circ\\:}+240^{\\circ\\:}\\right)=\\csc\\left(240^{\\circ\\:}\\right)$$" ], "result": "=\\csc\\left(240^{\\circ\\:}\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\csc\\left(240^{\\circ\\:}\\right)=-\\frac{2\\sqrt{3}}{3}$$", "input": "\\csc\\left(240^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\frac{1}{\\sin\\left(240^{\\circ\\:}\\right)}$$", "input": "\\csc\\left(240^{\\circ\\:}\\right)", "result": "=\\frac{1}{\\sin\\left(240^{\\circ\\:}\\right)}", "steps": [ { "type": "step", "primary": "Use the basic trigonometric identity: $$\\csc\\left(x\\right)=\\frac{1}{\\sin\\left(x\\right)}$$", "result": "=\\frac{1}{\\sin\\left(240^{\\circ\\:}\\right)}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(240^{\\circ\\:}\\right)=-\\frac{\\sqrt{3}}{2}$$", "input": "\\sin\\left(240^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(180^{\\circ\\:}\\right)\\cos\\left(60^{\\circ\\:}\\right)+\\cos\\left(180^{\\circ\\:}\\right)\\sin\\left(60^{\\circ\\:}\\right)$$", "input": "\\sin\\left(240^{\\circ\\:}\\right)", "result": "=\\sin\\left(180^{\\circ\\:}\\right)\\cos\\left(60^{\\circ\\:}\\right)+\\cos\\left(180^{\\circ\\:}\\right)\\sin\\left(60^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Write $$\\sin\\left(240^{\\circ\\:}\\right)\\:$$as $$\\sin\\left(180^{\\circ\\:}+60^{\\circ\\:}\\right)$$", "result": "=\\sin\\left(180^{\\circ\\:}+60^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Use the Angle Sum identity: $$\\sin\\left(s+t\\right)=\\sin\\left(s\\right)\\cos\\left(t\\right)+\\cos\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\sin\\left(180^{\\circ\\:}\\right)\\cos\\left(60^{\\circ\\:}\\right)+\\cos\\left(180^{\\circ\\:}\\right)\\sin\\left(60^{\\circ\\:}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(180^{\\circ\\:}\\right)=0$$", "input": "\\sin\\left(180^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(60^{\\circ\\:}\\right)=\\frac{1}{2}$$", "input": "\\cos\\left(60^{\\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{1}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(180^{\\circ\\:}\\right)=\\left(-1\\right)$$", "input": "\\cos\\left(180^{\\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": "=\\left(-1\\right)" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(60^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\sin\\left(60^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$360^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&180^{\\circ }&0\\\\\\hline 30^{\\circ }&\\frac{1}{2}&210^{\\circ }&-\\frac{1}{2}\\\\\\hline 45^{\\circ }&\\frac{\\sqrt{2}}{2}&225^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 60^{\\circ }&\\frac{\\sqrt{3}}{2}&240^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 90^{\\circ }&1&270^{\\circ }&-1\\\\\\hline 120^{\\circ }&\\frac{\\sqrt{3}}{2}&300^{\\circ }&-\\frac{\\sqrt{3}}{2}\\\\\\hline 135^{\\circ }&\\frac{\\sqrt{2}}{2}&315^{\\circ }&-\\frac{\\sqrt{2}}{2}\\\\\\hline 150^{\\circ }&\\frac{1}{2}&330^{\\circ }&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0\\cdot\\:\\frac{1}{2}+\\left(-1\\right)\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Simplify", "result": "=-\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "result": "=\\frac{1}{-\\frac{\\sqrt{3}}{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{-\\frac{\\sqrt{3}}{2}}:{\\quad}-\\frac{2\\sqrt{3}}{3}$$", "input": "\\frac{1}{-\\frac{\\sqrt{3}}{2}}", "result": "=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{1}{\\frac{\\sqrt{3}}{2}}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{1}{\\frac{b}{c}}=\\frac{c}{b}$$", "secondary": [ "$$\\frac{1}{\\frac{\\sqrt{3}}{2}}=\\frac{2}{\\sqrt{3}}$$" ], "result": "=-\\frac{2}{\\sqrt{3}}" }, { "type": "interim", "title": "Rationalize $$-\\frac{2}{\\sqrt{3}}:{\\quad}-\\frac{2\\sqrt{3}}{3}$$", "input": "-\\frac{2}{\\sqrt{3}}", "result": "=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=-\\frac{2\\sqrt{3}}{\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "interim", "title": "$$\\sqrt{3}\\sqrt{3}=3$$", "input": "\\sqrt{3}\\sqrt{3}", "result": "=-\\frac{2\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KTpmXttz9tzhFROsvdfdpl9t0rGzXfqIeCX9GJLyBFHMwViaLUXkeD+JukROhWdjqqRb86vK1jAPxwJmTBc7jxuwBfr4jqpPqXmFXXvxZ1uqe0B1E2GM2M0eTcXoyACf" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DnLLpBZXK8BfGeNnJwslXCcg0t9KufQ5/Ecxx5MB741FdiKL0bo+0lLRUzNXm1IEebTKSUwQXHFFpEsrzHtUqUYz5rCMohOVKo1AqkixmUCublrFVHaORbFnakMRHJmeixpjZyz1hn/Tx95aZo6CSFjbCAuoanhwjo/9bMp/72nXJXqehyuhyA5YIl5w+yTH9SCdniNpyTQQU7VSlVeZmw==" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "result": "=-\\left(-\\frac{2\\sqrt{3}}{3}\\right)" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{2\\sqrt{3}}{3}" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }