sin(pi/3-pi/4)

{ "query": { "display": "$$\\sin\\left(\\frac{π}{3}-\\frac{π}{4}\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sin(\\frac{π}{3}-\\frac{π}{4})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{\\sqrt{2}(\\sqrt{3}-1)}{4}", "decimal": "0.25881…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(\\frac{π}{3}-\\frac{π}{4}\\right)=\\frac{\\sqrt{2}\\left(\\sqrt{3}-1\\right)}{4}$$", "input": "\\sin\\left(\\frac{π}{3}-\\frac{π}{4}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\frac{π}{3}\\right)\\cos\\left(\\frac{π}{4}\\right)-\\cos\\left(\\frac{π}{3}\\right)\\sin\\left(\\frac{π}{4}\\right)$$", "input": "\\sin\\left(\\frac{π}{3}-\\frac{π}{4}\\right)", "result": "=\\sin\\left(\\frac{π}{3}\\right)\\cos\\left(\\frac{π}{4}\\right)-\\cos\\left(\\frac{π}{3}\\right)\\sin\\left(\\frac{π}{4}\\right)", "steps": [ { "type": "step", "primary": "Use the Angle Difference 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{π}{3}\\right)\\cos\\left(\\frac{π}{4}\\right)-\\cos\\left(\\frac{π}{3}\\right)\\sin\\left(\\frac{π}{4}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{π}{3}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\sin\\left(\\frac{π}{3}\\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{3}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value 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:$${\\quad}\\cos\\left(\\frac{π}{3}\\right)=\\frac{1}{2}$$", "input": "\\cos\\left(\\frac{π}{3}\\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{1}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{π}{4}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\sin\\left(\\frac{π}{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{\\sqrt{3}}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}-\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}" }, { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}-\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}:{\\quad}\\frac{\\sqrt{2}\\left(\\sqrt{3}-1\\right)}{4}$$", "input": "\\frac{\\sqrt{3}}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}-\\frac{1}{2}\\cdot\\:\\frac{\\sqrt{2}}{2}", "result": "=\\frac{\\sqrt{2}\\left(\\sqrt{3}-1\\right)}{4}", "steps": [ { "type": "step", "primary": "Factor out common term $$\\frac{\\sqrt{2}}{2}$$", "result": "=\\frac{\\sqrt{2}}{2}\\left(\\frac{\\sqrt{3}}{2}-\\frac{1}{2}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "$$\\frac{\\sqrt{3}}{2}-\\frac{1}{2}=\\frac{\\sqrt{3}-1}{2}$$", "input": "\\frac{\\sqrt{3}}{2}-\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\sqrt{3}-1}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMVF0iQ8664zDgtvyFD1NCE4nVQZ9jTQhWaoOGaFYelMKo5FYteSPKwXny4uCMrdsKyb97z4ciRBzgXtcnYl8cikTWVdsMGgVRGpt0sdO/xIsSgixzvqNSsyx877fl3aO/EsXiOOw6wqajw39NwQqTA8fGlIqBWAxUqVtmOOTYfFxgX5kbaH575RctZlINcWWDSS3daIZHtloJpe/PvtsyNI=" } }, { "type": "step", "result": "=\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}-1}{2}" }, { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{\\left(\\sqrt{3}-1\\right)\\sqrt{2}}{2\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=\\frac{\\sqrt{2}\\left(\\sqrt{3}-1\\right)}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMZDr7b2C7YrlcdHHbaZvc/exmKsnFhk/kBL6XsYsTqTto+oeMNyGiMYSzgI5Qndqbgmv19wq0Y68zdO7Y2e9yjfKF/s2NQUD3NRcmXAMVuy2A585Wz2Y8ioMtXlAhbC3efcFBBxtnU7tj9iPqSMJYMGEzp3RYdebOQ4wlsVSjzfCNr/ByswV6uvdLq/uE2jcE03kCh3oevUunZ7/b0qFKBSBAc1PafP4ia+acEW7bvr5asHWSnLvgfYYU0cg1ayHTQmv19wq0Y68zdO7Y2e9yjffDPb8uOqfMHoMNypYw2xsZe+3IobTh1qApFsHX0PdPLGYqycWGT+QEvpexixOpO2JqVxX90jlMfh9fKn6dzC4" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }