cos(255)

{ "query": { "display": "$$\\cos\\left(255^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\cos(255^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{\\sqrt{2}-\\sqrt{6}}{4}", "decimal": "-0.25881…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(255^{\\circ\\:}\\right)=\\frac{\\sqrt{2}-\\sqrt{6}}{4}$$", "input": "\\cos\\left(255^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(135^{\\circ\\:}\\right)\\cos\\left(120^{\\circ\\:}\\right)-\\sin\\left(135^{\\circ\\:}\\right)\\sin\\left(120^{\\circ\\:}\\right)$$", "input": "\\cos\\left(255^{\\circ\\:}\\right)", "result": "=\\cos\\left(135^{\\circ\\:}\\right)\\cos\\left(120^{\\circ\\:}\\right)-\\sin\\left(135^{\\circ\\:}\\right)\\sin\\left(120^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Write $$\\cos\\left(255^{\\circ\\:}\\right)\\:$$as $$\\cos\\left(135^{\\circ\\:}+120^{\\circ\\:}\\right)$$", "result": "=\\cos\\left(135^{\\circ\\:}+120^{\\circ\\:}\\right)" }, { "type": "step", "primary": "Use the Angle Sum identity: $$\\cos\\left(s+t\\right)=\\cos\\left(s\\right)\\cos\\left(t\\right)-\\sin\\left(s\\right)\\sin\\left(t\\right)$$", "result": "=\\cos\\left(135^{\\circ\\:}\\right)\\cos\\left(120^{\\circ\\:}\\right)-\\sin\\left(135^{\\circ\\:}\\right)\\sin\\left(120^{\\circ\\:}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(135^{\\circ\\:}\\right)=-\\frac{\\sqrt{2}}{2}$$", "input": "\\cos\\left(135^{\\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{2}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(120^{\\circ\\:}\\right)=-\\frac{1}{2}$$", "input": "\\cos\\left(120^{\\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:$${\\quad}\\sin\\left(135^{\\circ\\:}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\sin\\left(135^{\\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{2}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(120^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\sin\\left(120^{\\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": "=\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(-\\frac{1}{2}\\right)-\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Simplify $$\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(-\\frac{1}{2}\\right)-\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}:{\\quad}\\frac{\\sqrt{2}-\\sqrt{6}}{4}$$", "input": "\\left(-\\frac{\\sqrt{2}}{2}\\right)\\left(-\\frac{1}{2}\\right)-\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}", "result": "=\\frac{\\sqrt{2}-\\sqrt{6}}{4}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{1}{2}-\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "$$\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{1}{2}=\\frac{\\sqrt{2}}{4}$$", "input": "\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\sqrt{2}\\cdot\\:1}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply: $$\\sqrt{2}\\cdot\\:1=\\sqrt{2}$$", "result": "=\\frac{\\sqrt{2}}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{\\sqrt{2}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMb3At0hgR6LrRkI9bvhUawb8JGnBYXAaSTCT4nOaaDitA585Wz2Y8ioMtXlAhbC3efcFBBxtnU7tj9iPqSMJYMGKrPCBzH+7CEDRtwpnkmglzVshnF+h2bdSxlx9kBmvixqxDwJ92JV1QKn+tjCIbSGrr3fOnUl2JFm24EdyiS1w9i2TaxQ/bvHZVnGcgAbPeT8pgCUGPTBnYP7H4qH9EFM=" } }, { "type": "interim", "title": "$$\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}=\\frac{\\sqrt{6}}{4}$$", "input": "\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\sqrt{2}\\sqrt{3}}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{\\sqrt{2}\\sqrt{3}}{4}" }, { "type": "interim", "title": "Simplify $$\\sqrt{2}\\sqrt{3}:{\\quad}\\sqrt{6}$$", "input": "\\sqrt{2}\\sqrt{3}", "result": "=\\frac{\\sqrt{6}}{4}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{b}=\\sqrt{a\\cdot{b}}$$", "secondary": [ "$$\\sqrt{2}\\sqrt{3}=\\sqrt{2\\cdot\\:3}$$" ], "result": "=\\sqrt{2\\cdot\\:3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\sqrt{6}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMb3At0hgR6LrRkI9bvhUawYnINLfSrn0OfxHMceTAe+NfJEe4bDv60zkwRNMOPq9UULJUwysTkf+ifmThr1HGrp+VTqPi0iEFnY1TZ2BS4ZpcTZnm/8YWra/1y/CNJ8TTBwxYXwitbTZlgK2Hv6yLaS9wLdIYEei60ZCPW74VGsGJyDS30q59Dn8RzHHkwHvjfYtk2sUP27x2VZxnIAGz3navp7rMsdbVpq1Tk9ICmsH" } }, { "type": "step", "result": "=\\frac{\\sqrt{2}}{4}-\\frac{\\sqrt{6}}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\sqrt{2}-\\sqrt{6}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75s3xpOjgqYGSPN+T51/sNe8CHspmOsOW/MSUbIv35Pvb2yoeTfF1/796Lu/OYThgzurCwdaPaK36YTauhcWWRuYDuNPy03m5Ds4/0zuq//VQgPwnKT4P5jIr+96R/uURdYPfXQvX4/bINBB8wSEQ0ezNQwy21Qht4dZg+h1P+FAJC+8vYFyYMWoHOzI+asPb+yJwO2vX/hJkuWfCtbSyY+5AIz++qluupTlLFEcE9J0DWENzqsGATxIhXzclB1c3/myFPEJe25vrLXnBUG30XneINUzCJW1veCxuzR5+s1KxmKsnFhk/kBL6XsYsTqTtLp/Bib5CanrsDTuvzFW/0DWLn0soyX+mV3fHPQtIVEk=" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }