1/2 sin(2pi)

{ "query": { "display": "$$\\frac{1}{2}\\sin\\left(2π\\right)$$", "symbolab_question": "TRIG_EVALUATE#\\frac{1}{2}\\sin(2π)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{1}{2}\\sin\\left(2π\\right)=0$$", "input": "\\frac{1}{2}\\sin\\left(2π\\right)", "steps": [ { "type": "interim", "title": "Rewrite using trig identities:$${\\quad}\\sin\\left(2π\\right)=0$$", "input": "\\sin\\left(2π\\right)", "steps": [ { "type": "interim", "title": "$$\\sin\\left(2π\\right)=\\sin\\left(0\\right)$$", "input": "\\sin\\left(2π\\right)", "result": "=\\sin\\left(0\\right)", "steps": [ { "type": "step", "primary": "Rewrite $$2π$$ as $$2π+0$$", "result": "=\\sin\\left(2π+0\\right)" }, { "type": "step", "primary": "Apply the periodicity of $$\\sin$$: $$\\sin\\left(x+2π\\right)=\\sin\\left(x\\right)$$", "secondary": [ "$$\\sin\\left(2π+0\\right)=\\sin\\left(0\\right)$$" ], "result": "=\\sin\\left(0\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(0\\right)=0$$", "input": "\\sin\\left(0\\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": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities Title 0Eq" } }, { "type": "step", "result": "=\\frac{1}{2}\\cdot\\:0" }, { "type": "step", "primary": "Simplify", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }