-4cos(pi/3)

{ "query": { "display": "$$-4\\cos\\left(\\frac{π}{3}\\right)$$", "symbolab_question": "TRIG_EVALUATE#-4\\cos(\\frac{π}{3})" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-2", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-4\\cos\\left(\\frac{π}{3}\\right)=-2$$", "input": "-4\\cos\\left(\\frac{π}{3}\\right)", "steps": [ { "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": "step", "result": "=-4\\cdot\\:\\frac{1}{2}" }, { "type": "interim", "title": "Simplify $$-4\\cdot\\:\\frac{1}{2}:{\\quad}-2$$", "input": "-4\\cdot\\:\\frac{1}{2}", "result": "=-2", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$4=\\frac{4}{1}$$", "result": "=\\frac{4}{1}\\cdot\\:\\frac{1}{2}" }, { "type": "step", "primary": "Cross-cancel common factor: $$2$$", "result": "=-\\frac{2}{1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{1}=a$$", "secondary": [ "$$\\frac{2}{1}=2$$" ], "result": "=-2" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7d8MvEanxv2v9ZRaXRuEiR7CVpDAG11IdQLl4eKYIiFMJQJZuTAY5js+oqjdT8ksl2wi7sDFXUdWBamMMAmaYmx429vuTSxWa7B/X3D1oP00QUIwKJB72YuhKozX/HBYxQPpwPm1Z+hwEbINNneEzRyS3daIZHtloJpe/PvtsyNI=" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }