9sin(75)

{ "query": { "display": "$$9\\sin\\left(75^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#9\\sin(75^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{9(\\sqrt{6}+\\sqrt{2})}{4}", "decimal": "8.69333…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$9\\sin\\left(75^{\\circ\\:}\\right)=\\frac{9\\left(\\sqrt{6}+\\sqrt{2}\\right)}{4}$$", "input": "9\\sin\\left(75^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(75^{\\circ\\:}\\right)=\\frac{\\sqrt{6}+\\sqrt{2}}{4}$$", "input": "\\sin\\left(75^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(45^{\\circ\\:}\\right)\\cos\\left(30^{\\circ\\:}\\right)+\\cos\\left(45^{\\circ\\:}\\right)\\sin\\left(30^{\\circ\\:}\\right)$$", "input": "\\sin\\left(75^{\\circ\\:}\\right)", "result": "=\\sin\\left(45^{\\circ\\:}\\right)\\cos\\left(30^{\\circ\\:}\\right)+\\cos\\left(45^{\\circ\\:}\\right)\\sin\\left(30^{\\circ\\:}\\right)", "steps": [ { "type": "step", "primary": "Write $$\\sin\\left(75^{\\circ\\:}\\right)\\:$$as $$\\sin\\left(45^{\\circ\\:}+30^{\\circ\\:}\\right)$$", "result": "=\\sin\\left(45^{\\circ\\:}+30^{\\circ\\:}\\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(45^{\\circ\\:}\\right)\\cos\\left(30^{\\circ\\:}\\right)+\\cos\\left(45^{\\circ\\:}\\right)\\sin\\left(30^{\\circ\\:}\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "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": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(30^{\\circ\\:}\\right)=\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(30^{\\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{3}}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(45^{\\circ\\:}\\right)=\\frac{\\sqrt{2}}{2}$$", "input": "\\cos\\left(45^{\\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:$${\\quad}\\sin\\left(30^{\\circ\\:}\\right)=\\frac{1}{2}$$", "input": "\\sin\\left(30^{\\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{1}{2}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}+\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{1}{2}" }, { "type": "interim", "title": "Simplify $$\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}+\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{1}{2}:{\\quad}\\frac{\\sqrt{6}+\\sqrt{2}}{4}$$", "input": "\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{\\sqrt{3}}{2}+\\frac{\\sqrt{2}}{2}\\cdot\\:\\frac{1}{2}", "result": "=\\frac{\\sqrt{6}+\\sqrt{2}}{4}", "steps": [ { "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": "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": "step", "result": "=\\frac{\\sqrt{6}}{4}+\\frac{\\sqrt{2}}{4}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\sqrt{6}+\\sqrt{2}}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74DwjiBDVBIb8+yh6fkkMMb3At0hgR6LrRkI9bvhUawYnINLfSrn0OfxHMceTAe+NfAXnCzCmfdPlff9vTGOwir3At0hgR6LrRkI9bvhUawb8JGnBYXAaSTCT4nOaaDitA585Wz2Y8ioMtXlAhbC3efcFBBxtnU7tj9iPqSMJYMEhEJHE+VZsy3z9vilJqnGjU7VPSycU4+opcv5LozFWaZjcBIL5pmo83UMFZRSzJsUt8Vz4ri5I/oyZLRAY9v9SNVXuPxzUYu+xFX0tWjdAia3Yn1NFotSssbSwtvx+GFYdOivu9vLUN4LqsEmHHs7TNVXuPxzUYu+xFX0tWjdAiUD6cD5tWfocBGyDTZ3hM0ckt3WiGR7ZaCaXvz77bMjS" } } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "result": "=9\\cdot\\:\\frac{\\sqrt{6}+\\sqrt{2}}{4}" }, { "type": "interim", "title": "Simplify $$9\\cdot\\:\\frac{\\sqrt{6}+\\sqrt{2}}{4}:{\\quad}\\frac{9\\left(\\sqrt{6}+\\sqrt{2}\\right)}{4}$$", "input": "9\\cdot\\:\\frac{\\sqrt{6}+\\sqrt{2}}{4}", "result": "=\\frac{9\\left(\\sqrt{6}+\\sqrt{2}\\right)}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(\\sqrt{6}+\\sqrt{2}\\right)\\cdot\\:9}{4}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tCaovvy4r4yHTcJnXLMZMNwiMfXGqdsuFN6iueZaypx8tiOWRnAJ00H0w37QNxWhdYPfXQvX4/bINBB8wSEQ0QSS8+xqOEyFxV0BPkbhPgUhEJHE+VZsy3z9vilJqnGjA9gyHrG+1IfKIV5zoAUeT0UqTd96MWTKI6Kr2Ib0iQAi5KYlQO0vFE/Inns2SruqfhLTFwr36gYOS56QG86uXnguC/sjoQb/Xb/A4POjPG8/KYAlBj0wZ2D+x+Kh/RBT" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }