sin(19/12 pi)

{ "query": { "display": "$$\\sin\\left(\\frac{19}{12}π\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sin(\\frac{19}{12}π)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{\\sqrt{2}(-1-\\sqrt{3})}{4}", "decimal": "-0.96592…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(\\frac{19}{12}π\\right)=\\frac{\\sqrt{2}\\left(-1-\\sqrt{3}\\right)}{4}$$", "input": "\\sin\\left(\\frac{19}{12}π\\right)", "steps": [ { "type": "interim", "title": "Simplify:$${\\quad}\\frac{19}{12}π=\\frac{19π}{12}$$", "input": "\\frac{19}{12}π", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{19π}{12}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "=\\sin\\left(\\frac{19π}{12}\\right)" }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\frac{5π}{6}\\right)\\cos\\left(\\frac{3π}{4}\\right)+\\cos\\left(\\frac{5π}{6}\\right)\\sin\\left(\\frac{3π}{4}\\right)$$", "input": "\\sin\\left(\\frac{19π}{12}\\right)", "result": "=\\sin\\left(\\frac{5π}{6}\\right)\\cos\\left(\\frac{3π}{4}\\right)+\\cos\\left(\\frac{5π}{6}\\right)\\sin\\left(\\frac{3π}{4}\\right)", "steps": [ { "type": "step", "primary": "Write $$\\sin\\left(\\frac{19π}{12}\\right)\\:$$as $$\\sin\\left(\\frac{5π}{6}+\\frac{3π}{4}\\right)$$", "result": "=\\sin\\left(\\frac{5π}{6}+\\frac{3π}{4}\\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(\\frac{5π}{6}\\right)\\cos\\left(\\frac{3π}{4}\\right)+\\cos\\left(\\frac{5π}{6}\\right)\\sin\\left(\\frac{3π}{4}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{5π}{6}\\right)=\\frac{1}{2}$$", "input": "\\sin\\left(\\frac{5π}{6}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{1}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(\\frac{3π}{4}\\right)=-\\frac{\\sqrt{2}}{2}$$", "input": "\\cos\\left(\\frac{3π}{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:$${\\quad}\\cos\\left(\\frac{5π}{6}\\right)=-\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(\\frac{5π}{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": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{3π}{4}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\sin\\left(\\frac{3π}{4}\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{2}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{1}{2}\\left(-\\frac{\\sqrt{2}}{2}\\right)+\\left(-\\frac{\\sqrt{3}}{2}\\right)\\frac{\\sqrt{2}}{2}" }, { "type": "interim", "title": "Simplify $$\\frac{1}{2}\\left(-\\frac{\\sqrt{2}}{2}\\right)+\\left(-\\frac{\\sqrt{3}}{2}\\right)\\frac{\\sqrt{2}}{2}:{\\quad}\\frac{\\sqrt{2}\\left(-1-\\sqrt{3}\\right)}{4}$$", "input": "\\frac{1}{2}\\left(-\\frac{\\sqrt{2}}{2}\\right)+\\left(-\\frac{\\sqrt{3}}{2}\\right)\\frac{\\sqrt{2}}{2}", "result": "=\\frac{\\sqrt{2}\\left(-1-\\sqrt{3}\\right)}{4}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}-\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}" }, { "type": "step", "primary": "Factor out common term $$\\frac{\\sqrt{2}}{2}$$", "result": "=\\frac{\\sqrt{2}}{2}\\left(-\\frac{1}{2}-\\frac{\\sqrt{3}}{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "$$-\\frac{1}{2}-\\frac{\\sqrt{3}}{2}=\\frac{-1-\\sqrt{3}}{2}$$", "input": "-\\frac{1}{2}-\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{-1-\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70akGKguCyl0JWkEcJv5brSb97z4ciRBzgXtcnYl8cimdFkLz9sJGikOYQponj2jhzMFYmi1F5Hg/ibpEToVnY7yGV2ksBzAuSkEiOxIy75utCZA9IvYdgMxbjpdehNrsQMzYPVCoWxv5R3Rvt7yY2xl+UX3czAd4J9AmAwqpKbJe/LQerB/1nXBBlKpYAjZdx21eiv0GO3vgMtTZ/oEoNo8BPOx0wlsgFN8qUa6AzA0=" } }, { "type": "step", "result": "=\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{-1-\\sqrt{3}}{2}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\left(-1-\\sqrt{3}\\right)\\sqrt{2}}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{\\sqrt{2}\\left(-1-\\sqrt{3}\\right)}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79jHpDL2BX3rbBpw0SEa3ii6yyeeeKwmd0ICbjP7OejwlzSNbmbXK3EGl01gn+6hu/VB52xBSkwRjOQwSfivB77GYqycWGT+QEvpexixOpO18kR7hsO/rTOTBE0w4+r1RQslTDKxOR/6J+ZOGvUcaupQy+ZBd9sMgZk15/ZoDWNQ8/6qa5tQvMY98SH1laUxGP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXSTlmEIIkihPa/D/9QBZWwDeMHJjS1WbRpXjhL2kWYmx0monWTWu08OF4U0fcjTgWoePbL1EZJe2lsl/GJz9E+jAEDaREwoEOwO7gdh5V4G6I8BPOx0wlsgFN8qUa6AzA0=" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }