cos((5pi)/(12))+cos(pi/(12))

{ "query": { "display": "$$\\cos\\left(\\frac{5π}{12}\\right)+\\cos\\left(\\frac{π}{12}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\cos(\\frac{5π}{12})+\\cos(\\frac{π}{12})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\sqrt{\\frac{3}{2}}", "decimal": "1.22474…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(\\frac{5π}{12}\\right)+\\cos\\left(\\frac{π}{12}\\right)=\\sqrt{\\frac{3}{2}}$$", "input": "\\cos\\left(\\frac{5π}{12}\\right)+\\cos\\left(\\frac{π}{12}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}2\\cos\\left(\\frac{π}{4}\\right)\\cos\\left(\\frac{π}{6}\\right)$$", "input": "\\cos\\left(\\frac{5π}{12}\\right)+\\cos\\left(\\frac{π}{12}\\right)", "result": "=2\\cos\\left(\\frac{π}{4}\\right)\\cos\\left(\\frac{π}{6}\\right)", "steps": [ { "type": "step", "primary": "Use the Sum to Product identity: $$\\cos\\left(s\\right)+\\cos\\left(t\\right)=2\\cos\\left(\\frac{s+t}{2}\\right)\\cos\\left(\\frac{s-t}{2}\\right)$$", "result": "=2\\cos\\left(\\frac{\\frac{5π}{12}+\\frac{π}{12}}{2}\\right)\\cos\\left(\\frac{\\frac{5π}{12}-\\frac{π}{12}}{2}\\right)" }, { "type": "interim", "title": "Simplify:$${\\quad}\\frac{\\frac{5π}{12}+\\frac{π}{12}}{2}=\\frac{π}{4}$$", "input": "\\frac{\\frac{5π}{12}+\\frac{π}{12}}{2}", "steps": [ { "type": "interim", "title": "Combine the fractions $$\\frac{5π}{12}+\\frac{π}{12}:{\\quad}\\frac{π}{2}$$", "result": "=\\frac{\\frac{π}{2}}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{5π+π}{12}" }, { "type": "step", "primary": "Add similar elements: $$5π+π=6π$$", "result": "=\\frac{6π}{12}" }, { "type": "step", "primary": "Cancel the common factor: $$6$$", "result": "=\\frac{π}{2}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{π}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "interim", "title": "Simplify:$${\\quad}\\frac{\\frac{5π}{12}-\\frac{π}{12}}{2}=\\frac{π}{6}$$", "input": "\\frac{\\frac{5π}{12}-\\frac{π}{12}}{2}", "steps": [ { "type": "interim", "title": "Combine the fractions $$\\frac{5π}{12}-\\frac{π}{12}:{\\quad}\\frac{π}{3}$$", "result": "=\\frac{\\frac{π}{3}}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{5π-π}{12}" }, { "type": "step", "primary": "Add similar elements: $$5π-π=4π$$", "result": "=\\frac{4π}{12}" }, { "type": "step", "primary": "Cancel the common factor: $$4$$", "result": "=\\frac{π}{3}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{3\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=\\frac{π}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "=2\\cos\\left(\\frac{π}{4}\\right)\\cos\\left(\\frac{π}{6}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(\\frac{π}{4}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\cos\\left(\\frac{π}{4}\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(\\frac{π}{6}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(\\frac{π}{6}\\right)", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}:{\\quad}\\sqrt{\\frac{3}{2}}$$", "input": "2\\cdot\\:\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}", "result": "=\\sqrt{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}\\cdot\\frac{d}{e}=\\frac{a\\:\\cdot\\:b\\:\\cdot\\:d}{c\\:\\cdot\\:e}$$", "result": "=\\frac{\\sqrt{2}\\sqrt{3}\\cdot\\:2}{2\\cdot\\:2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=\\frac{\\sqrt{2}\\sqrt{3}}{2}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{\\frac{1}{2}}\\sqrt{3}}{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{2^{\\frac{1}{2}}}{2^{1}}=\\frac{1}{2^{1-\\frac{1}{2}}}$$" ], "result": "=\\frac{\\sqrt{3}}{2^{1-\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=\\frac{\\sqrt{3}}{2^{\\frac{1}{2}}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$2^{\\frac{1}{2}}=\\sqrt{2}$$" ], "result": "=\\frac{\\sqrt{3}}{\\sqrt{2}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Combine same powers : $$\\frac{\\sqrt{x}}{\\sqrt{y}}=\\sqrt{\\frac{x}{y}}$$", "result": "=\\sqrt{\\frac{3}{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvihqxDwJ92JV1QKn+tjCIbSEJr9fcKtGOvM3Tu2Nnvco3iJ7qJvbOf36tmN4DMHd31QOfOVs9mPIqDLV5QIWwt3kULfQ2PlgyGQK03mK0UmF0E1J+51NQ1TnXtcinw7AvffC30sSftAIFS6Qkpy19Ikp0x8irWzKvvppKMUvqTrnrCa/X3CrRjrzN07tjZ73KNw8dCbJqMwzc7hfouUCYDF8m/e8+HIkQc4F7XJ2JfHIpbqwnNL+vkns16SbvknXX+A==" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }