[2(cos(pi/3)+isin(pi/3))]^4

{ "query": { "display": "$$[2\\left(\\cos\\left(\\frac{π}{3}\\right)+i\\sin\\left(\\frac{π}{3}\\right)\\right)]^{4}$$", "symbolab_question": "TRIG_EVALUATE#[2(\\cos(\\frac{π}{3})+i\\sin(\\frac{π}{3}))]^{4}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "(1+\\sqrt{3}i)^{4}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$[2\\left(\\cos\\left(\\frac{π}{3}\\right)+i\\sin\\left(\\frac{π}{3}\\right)\\right)]^{4}=\\left(1+\\sqrt{3}i\\right)^{4}$$", "input": "[2\\left(\\cos\\left(\\frac{π}{3}\\right)+i\\sin\\left(\\frac{π}{3}\\right)\\right)]^{4}", "steps": [ { "type": "step", "result": "=\\left(2\\left(\\cos\\left(\\frac{π}{3}\\right)+i\\sin\\left(\\frac{π}{3}\\right)\\right)\\right)^{4}" }, { "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:$${\\quad}\\sin\\left(\\frac{π}{3}\\right)=\\frac{\\sqrt{3}}{2}$$", "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": "step", "result": "=\\left(2\\left(\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}\\right)\\right)^{4}" }, { "type": "interim", "title": "Simplify $$\\left(2\\left(\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}\\right)\\right)^{4}:{\\quad}\\left(1+\\sqrt{3}i\\right)^{4}$$", "input": "\\left(2\\left(\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}\\right)\\right)^{4}", "result": "=\\left(1+\\sqrt{3}i\\right)^{4}", "steps": [ { "type": "interim", "title": "Multiply $$i\\frac{\\sqrt{3}}{2}\\::{\\quad}\\frac{\\sqrt{3}i}{2}$$", "input": "i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\sqrt{3}i}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "step", "result": "=\\left(2\\left(\\frac{\\sqrt{3}i}{2}+\\frac{1}{2}\\right)\\right)^{4}" }, { "type": "interim", "title": "Combine the fractions $$\\frac{1}{2}+\\frac{\\sqrt{3}i}{2}:{\\quad}\\frac{1+\\sqrt{3}i}{2}$$", "result": "=\\left(2\\left(\\frac{1+\\sqrt{3}i}{2}\\right)\\right)^{4}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{1+\\sqrt{3}i}{2}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\left(2\\cdot\\:\\frac{1+\\sqrt{3}i}{2}\\right)^{4}" }, { "type": "interim", "title": "Multiply $$2\\cdot\\:\\frac{1+\\sqrt{3}i}{2}\\::{\\quad}1+\\sqrt{3}i$$", "input": "2\\cdot\\:\\frac{1+\\sqrt{3}i}{2}", "result": "=\\left(1+\\sqrt{3}i\\right)^{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{\\left(1+\\sqrt{3}i\\right)\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1+\\sqrt{3}i" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7U9WHxgAszjLcGmIO22MYmuSrYp04Nbz4i1CtP/p1cV9i+DuRWrfYB0PuPFtw4Rg13oZCq59Hq2va8/E5S/sf77D6EH27/LwaegRKJptcxw+RxIwAB55K1E33KF+B0d4X72wZm7kDUxdE6YSmfEbr2tZjUi/hD1FOWV6wYZcR2lgl3v7EjvvcM4bDzEGoIputJv3vPhyJEHOBe1ydiXxyKa4cxnhbIHGHyPrbriTKbPY=" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }