sin(2arctan(4))

{ "query": { "display": "$$\\sin\\left(2\\arctan\\left(4\\right)\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\sin(2\\arctan(4))" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "\\frac{8}{17}", "decimal": "0.47058…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(2\\arctan\\left(4\\right)\\right)=\\frac{8}{17}$$", "input": "\\sin\\left(2\\arctan\\left(4\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}2\\sin\\left(\\arctan\\left(4\\right)\\right)\\cos\\left(\\arctan\\left(4\\right)\\right)$$", "input": "\\sin\\left(2\\arctan\\left(4\\right)\\right)", "result": "=2\\sin\\left(\\arctan\\left(4\\right)\\right)\\cos\\left(\\arctan\\left(4\\right)\\right)", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$\\sin\\left(2x\\right)=2\\sin\\left(x\\right)\\cos\\left(x\\right)$$", "result": "=2\\sin\\left(\\arctan\\left(4\\right)\\right)\\cos\\left(\\arctan\\left(4\\right)\\right)" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\arctan\\left(4\\right)\\right)=\\frac{4\\sqrt{17}}{17}$$", "input": "\\sin\\left(\\arctan\\left(4\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(\\arctan\\left(4\\right)\\right)=\\frac{4\\sqrt{1+4^{2}}}{1+4^{2}}$$", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\sin\\left(\\arctan\\left(x\\right)\\right)=\\frac{x\\sqrt{1+x^{2}}}{1+x^{2}}$$" }, { "type": "step", "result": "=\\frac{4\\sqrt{1+4^{2}}}{1+4^{2}}" } ], "meta": { "interimType": "Trig Use Identity 1Eq" } }, { "type": "step", "result": "=\\frac{4\\sqrt{1+4^{2}}}{1+4^{2}}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{4\\sqrt{17}}{17}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(\\arctan\\left(4\\right)\\right)=\\frac{\\sqrt{17}}{17}$$", "input": "\\cos\\left(\\arctan\\left(4\\right)\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\cos\\left(\\arctan\\left(4\\right)\\right)=\\frac{\\sqrt{1+4^{2}}}{1+4^{2}}$$", "steps": [ { "type": "step", "primary": "Use the following identity: $$\\cos\\left(\\arctan\\left(x\\right)\\right)=\\frac{\\sqrt{1+x^{2}}}{1+x^{2}}$$" }, { "type": "step", "result": "=\\frac{\\sqrt{1+4^{2}}}{1+4^{2}}" } ], "meta": { "interimType": "Trig Use Identity 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{1+4^{2}}}{1+4^{2}}" }, { "type": "step", "primary": "Simplify", "result": "=\\frac{\\sqrt{17}}{17}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "result": "=2\\cdot\\:\\frac{4\\sqrt{17}}{17}\\cdot\\:\\frac{\\sqrt{17}}{17}" }, { "type": "interim", "title": "Simplify $$2\\cdot\\:\\frac{4\\sqrt{17}}{17}\\cdot\\:\\frac{\\sqrt{17}}{17}:{\\quad}\\frac{8}{17}$$", "input": "2\\cdot\\:\\frac{4\\sqrt{17}}{17}\\cdot\\:\\frac{\\sqrt{17}}{17}", "result": "=\\frac{8}{17}", "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{4\\sqrt{17}\\sqrt{17}\\cdot\\:2}{17\\cdot\\:17}" }, { "type": "interim", "title": "$$4\\sqrt{17}\\sqrt{17}\\cdot\\:2=136$$", "input": "4\\sqrt{17}\\sqrt{17}\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=8\\sqrt{17}\\sqrt{17}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{17}\\sqrt{17}=17$$" ], "result": "=8\\cdot\\:17", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:17=136$$", "result": "=136" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7xAQ3AlOdwiNuw7uZ0gzt6/5L6KhuISkVhty0kMChHl4tOtZYwUjyXhDTsNnn6ElrvcIqdLaqXuCdSt1uG9+WLx0R3c0uYNhyRWpx951US4gVNdLIp7jabNu4Dg6mdZQOwt5EE8TDd/nQg9YvEE0lh7CI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\frac{136}{17\\cdot\\:17}" }, { "type": "step", "primary": "Multiply the numbers: $$17\\cdot\\:17=289$$", "result": "=\\frac{136}{289}" }, { "type": "step", "primary": "Cancel the common factor: $$17$$", "result": "=\\frac{8}{17}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/OsC643lXZbU+VEjF1qvigdyCMsLR8wolnNzIH5TPocun8GJvkJqeuwNO6/MVb/Qqmm5kve0Fk/rlxRmcykJbVXTSum/z5kLpMzXS1UJIezTENL5jlqPSGX4Xou8NoCWToaH8WVylhdgUF1T1BFHXnql8XXPq6bNQlMm+36iNhlr5cipPv6wYgg41dg0sAIkT2DdMmdQE/8Os4qb2jHbTiJAFFUY/6dLX4Z0swfCceldWuqgN8TweTVxItPgzZPjTUYVLqML6uRiDvTqu8pDOg==" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }