30cos(150)

{ "query": { "display": "$$30\\cos\\left(150^{\\circ\\:}\\right)$$", "symbolab_question": "TRIG_EVALUATE#30\\cos(150^{\\circ })" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Evaluate Functions", "subTopic": "Simplified", "default": "-15\\sqrt{3}", "decimal": "-25.98076…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$30\\cos\\left(150^{\\circ\\:}\\right)=-15\\sqrt{3}$$", "input": "30\\cos\\left(150^{\\circ\\:}\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\cos\\left(150^{\\circ\\:}\\right)=-\\frac{\\sqrt{3}}{2}$$", "input": "\\cos\\left(150^{\\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": "step", "result": "=30\\left(-\\frac{\\sqrt{3}}{2}\\right)" }, { "type": "interim", "title": "Simplify $$30\\left(-\\frac{\\sqrt{3}}{2}\\right):{\\quad}-15\\sqrt{3}$$", "input": "30\\left(-\\frac{\\sqrt{3}}{2}\\right)", "result": "=-15\\sqrt{3}", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=-30\\cdot\\:\\frac{\\sqrt{3}}{2}" }, { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=-\\frac{\\sqrt{3}\\cdot\\:30}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{30}{2}=15$$", "result": "=-15\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7JtgOVu5faxKN1/Fi85yxld0LoaIFsS7BOMSn94TTvSHdd47a0hQ8flDbGsI5To1dRYyuravk8vxB+pqZkLjzOqYo6P+y4hL8YqgVsNaOrT6Y3ASC+aZqPN1DBWUUsybF+yWs0bOvqvVjMyPZ8NaB6Ej+eLm1Sdr+9klmhmQtfjWwiNrEngO+NNvZ9sqNu+2V" } } ], "meta": { "solvingClass": "Trig Evaluate", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Evaluate%20Functions", "practiceTopic": "Evaluate Functions" } }, "meta": { "showVerify": true } }