(10)/(sin(45))

{ "query": { "display": "$$\\frac{10}{\\sin\\left(45^{\\circ\\:}\\right)}$$", "symbolab_question": "TRIG_EVALUATE#\\frac{10}{\\sin(45^{\\circ })}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "10\\sqrt{2}", "decimal": "14.14213…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{10}{\\sin\\left(45^{\\circ\\:}\\right)}=10\\sqrt{2}$$", "input": "\\frac{10}{\\sin\\left(45^{\\circ\\:}\\right)}", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(45^{\\circ\\:}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\sin\\left(45^{\\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": "step", "result": "=\\frac{10}{\\frac{\\sqrt{2}}{2}}" }, { "type": "interim", "title": "Simplify $$\\frac{10}{\\frac{\\sqrt{2}}{2}}:{\\quad}10\\sqrt{2}$$", "input": "\\frac{10}{\\frac{\\sqrt{2}}{2}}", "result": "=10\\sqrt{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$", "result": "=\\frac{10\\cdot\\:2}{\\sqrt{2}}" }, { "type": "step", "primary": "Multiply the numbers: $$10\\cdot\\:2=20$$", "result": "=\\frac{20}{\\sqrt{2}}" }, { "type": "interim", "title": "Factor $$20:{\\quad}2^{2}\\cdot\\:5$$", "steps": [ { "type": "step", "primary": "Factor $$20=2^{2}\\cdot\\:5$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}\\cdot\\:5}{\\sqrt{2}}" }, { "type": "interim", "title": "Cancel $$\\frac{2^{2}\\cdot\\:5}{\\sqrt{2}}:{\\quad}5\\cdot\\:2^{\\frac{3}{2}}$$", "input": "\\frac{2^{2}\\cdot\\:5}{\\sqrt{2}}", "result": "=5\\cdot\\:2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{2}=2^{\\frac{1}{2}}$$" ], "result": "=\\frac{2^{2}\\cdot\\:5}{2^{\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=x^{a-b}$$", "secondary": [ "$$\\frac{2^{2}}{2^{\\frac{1}{2}}}=2^{2-\\frac{1}{2}}$$" ], "result": "=5\\cdot\\:2^{-\\frac{1}{2}+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$2-\\frac{1}{2}=\\frac{3}{2}$$", "result": "=5\\cdot\\:2^{\\frac{3}{2}}" } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg/lyZ9NfP+aV8aypTubglP1vL2n70Edu1EmDkSCux/XzRqDxPUzBN6vjj5oJL9kUEFIXcrDCWaxGApXB2mHoHNBPo/eFeD8dSXXEaoeFmJ7RSpN33oxZMojoqvYhvSJAPqHzb2qXbidVuMnYVfxr8x9uOcVlUlPbWiUXl1uBs5XSi9QMWRpFnxTBB/oiH0ICCS3daIZHtloJpe/PvtsyNI=" } }, { "type": "interim", "title": "$$2^{\\frac{3}{2}}=2\\sqrt{2}$$", "input": "2^{\\frac{3}{2}}", "steps": [ { "type": "step", "primary": "$$2^{\\frac{3}{2}}=2^{1+\\frac{1}{2}}$$", "result": "=2^{1+\\frac{1}{2}}" }, { "type": "step", "primary": "Apply exponent rule: $$x^{a+b}=x^{a}x^{b}$$", "result": "=2^{1}\\cdot\\:2^{\\frac{1}{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=5\\cdot\\:2\\sqrt{2}" }, { "type": "step", "primary": "Multiply the numbers: $$5\\cdot\\:2=10$$", "result": "=10\\sqrt{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TMnmxy6yOVsSWvTekq/tV7GYqycWGT+QEvpexixOpO1FdiKL0bo+0lLRUzNXm1IEY2azjPkPCRG2cjooPAbcskdDQ8R/TjChY0JlkxjRMtbwt9LEn7QCBUukJKctfSJKm5KWrv5y7TlmVxX/22cMl5Cwpwxre5XHoP4zoc0XLIUNJFe8fE2e2rihldaPa0c9" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }