cos(θ)=(202)/(sqrt(331)*2\sqrt{33)}

{ "query": { "display": "$$\\cos\\left(θ\\right)=\\frac{202}{\\sqrt{331}\\cdot\\:2\\sqrt{33}}$$", "symbolab_question": "EQUATION#\\cos(θ)=\\frac{202}{\\sqrt{331}\\cdot 2\\sqrt{33}}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=0.26001…+2πn,θ=2π-0.26001…+2πn", "radians": "θ=0.26001…+6.28318…n,θ=6.02316…+6.28318…n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(θ\\right)=\\frac{202}{\\sqrt{331}\\cdot\\:2\\sqrt{33}}{\\quad:\\quad}θ=0.26001…+2πn,\\:θ=2π-0.26001…+2πn$$", "input": "\\cos\\left(θ\\right)=\\frac{202}{\\sqrt{331}\\cdot\\:2\\sqrt{33}}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{202}{\\sqrt{331}\\cdot\\:2\\sqrt{33}}:{\\quad}\\frac{101\\sqrt{10923}}{10923}$$", "input": "\\frac{202}{\\sqrt{331}\\cdot\\:2\\sqrt{33}}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{202}{2}=101$$", "result": "=\\frac{101}{\\sqrt{331}\\sqrt{33}}" }, { "type": "interim", "title": "Simplify $$\\sqrt{331}\\sqrt{33}:{\\quad}\\sqrt{10923}$$", "input": "\\sqrt{331}\\sqrt{33}", "result": "=\\frac{101}{\\sqrt{10923}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{b}=\\sqrt{a\\cdot{b}}$$", "secondary": [ "$$\\sqrt{331}\\sqrt{33}=\\sqrt{331\\cdot\\:33}$$" ], "result": "=\\sqrt{331\\cdot\\:33}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$331\\cdot\\:33=10923$$", "result": "=\\sqrt{10923}" } ], "meta": { "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Rationalize $$\\frac{101}{\\sqrt{10923}}:{\\quad}\\frac{101\\sqrt{10923}}{10923}$$", "input": "\\frac{101}{\\sqrt{10923}}", "result": "=\\frac{101\\sqrt{10923}}{10923}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{10923}}{\\sqrt{10923}}$$", "result": "=\\frac{101\\sqrt{10923}}{\\sqrt{10923}\\sqrt{10923}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{10923}$$" } } }, { "type": "interim", "title": "$$\\sqrt{10923}\\sqrt{10923}=10923$$", "input": "\\sqrt{10923}\\sqrt{10923}", "result": "=\\frac{101\\sqrt{10923}}{10923}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{10923}\\sqrt{10923}=10923$$" ], "result": "=10923", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7PaxoEANewcE4FPF0I1k6OTTxtUBAB/YBJaoPbX8xkoQgJ/ZZA32ZInFBpDtxBfiKLa73SUrYfPwjqAcDulvxNaN6Hv6MoTMtvtU0IQwXdn+71zjrUMZCO0dQJh6eIq0r1vuUEZpCDjFFdI1eHtWFGiS3daIZHtloJpe/PvtsyNI=" } } ], "meta": { "interimType": "Rationalize Title 1Eq" } }, { "type": "step", "result": "\\cos\\left(θ\\right)=\\frac{101\\sqrt{10923}}{10923}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wWAAnuVfCKBC6Nm6V0iJHeBhiOO6n3iAuxtTOg47mzqodJWUaaQlXZCAEzUvr3+4ICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHydczrLUwKm3qOS2NEbXDptQl41biKR29l5JYH1Mhfx9R3ql8XXPq6bNQlMm+36iNhljgtURsNZ8mF2q2lQDr86HJMGMvIN3YxCA4BOVbdb2ep6kb3UzqLz9+db3d/kAdb9R0uFnQQiSA1pL9yX8Qmo5" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(θ\\right)=\\frac{101\\sqrt{10923}}{10923}", "result": "θ=\\arccos\\left(\\frac{101\\sqrt{10923}}{10923}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{101\\sqrt{10923}}{10923}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(θ\\right)=\\frac{101\\sqrt{10923}}{10923}$$", "secondary": [ "$$\\cos\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(a\\right)+2πn,\\:\\quad\\:x=2π-\\arccos\\left(a\\right)+2πn$$" ], "result": "θ=\\arccos\\left(\\frac{101\\sqrt{10923}}{10923}\\right)+2πn,\\:θ=2π-\\arccos\\left(\\frac{101\\sqrt{10923}}{10923}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Show solutions in decimal form", "result": "θ=0.26001…+2πn,\\:θ=2π-0.26001…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\cos(θ)-\\frac{202}{\\sqrt{331}\\cdot 2\\sqrt{33}}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }