tan(2θ)=1,0<= θ<= 360

{ "query": { "display": "$$\\tan\\left(2θ\\right)=1,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}$$", "symbolab_question": "EQUATION#\\tan(2θ)=1,0^{\\circ }\\le θ\\le 360^{\\circ }" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=22.5^{\\circ },θ=112.5^{\\circ },θ=202.5^{\\circ },θ=292.5^{\\circ }", "radians": "θ=\\frac{π}{8},θ=\\frac{5π}{8},θ=\\frac{9π}{8},θ=\\frac{13π}{8}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan\\left(2θ\\right)=1,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}{\\quad:\\quad}θ=22.5^{\\circ\\:},\\:θ=112.5^{\\circ\\:},\\:θ=202.5^{\\circ\\:},\\:θ=292.5^{\\circ\\:}$$", "input": "\\tan\\left(2θ\\right)=1,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(2θ\\right)=1$$", "result": "2θ=45^{\\circ\\:}+180^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$180^{\\circ\\:}n$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline 30^{\\circ }&\\frac{\\sqrt{3}}{3}\\\\\\hline 45^{\\circ }&1\\\\\\hline 60^{\\circ }&\\sqrt{3}\\\\\\hline 90^{\\circ }&\\pm\\infty\\\\\\hline 120^{\\circ }&-\\sqrt{3}\\\\\\hline 135^{\\circ }&-1\\\\\\hline 150^{\\circ }&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "2θ=45^{\\circ\\:}+180^{\\circ\\:}n" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$2θ=45^{\\circ\\:}+180^{\\circ\\:}n:{\\quad}θ=22.5^{\\circ\\:}+\\frac{180^{\\circ\\:}n}{2}$$", "input": "2θ=45^{\\circ\\:}+180^{\\circ\\:}n", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2θ=45^{\\circ\\:}+180^{\\circ\\:}n", "result": "θ=22.5^{\\circ\\:}+\\frac{180^{\\circ\\:}n}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2θ}{2}=\\frac{45^{\\circ\\:}}{2}+\\frac{180^{\\circ\\:}n}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2θ}{2}=\\frac{45^{\\circ\\:}}{2}+\\frac{180^{\\circ\\:}n}{2}", "result": "θ=22.5^{\\circ\\:}+\\frac{180^{\\circ\\:}n}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2θ}{2}:{\\quad}θ$$", "input": "\\frac{2θ}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=θ" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7diSwzpX9ldlv2DLvSBEhWwCWKUbvV6WK3fDUgFtg3Q+kEtLScKDV+HqoDFDPEKGuZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz0ksNded2fMa8XATFfdmqDhsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Simplify $$\\frac{45^{\\circ\\:}}{2}+\\frac{180^{\\circ\\:}n}{2}:{\\quad}22.5^{\\circ\\:}+\\frac{180^{\\circ\\:}n}{2}$$", "input": "\\frac{45^{\\circ\\:}}{2}+\\frac{180^{\\circ\\:}n}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{45^{\\circ\\:}}{2}=22.5^{\\circ\\:}$$", "input": "\\frac{45^{\\circ\\:}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{180^{\\circ\\:}}{4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=22.5^{\\circ\\:}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OYxlL+/ci3l7qbr2GAQpOx1HTvsiFvlhwGizLG/6vzADnzlbPZjyKgy1eUCFsLd5d2wwppAqzHqmcVfO7CEUr10KrmheC67uOQV7VsC3iGcz0df5GrUs5IyqS4gC2nsrm+DSCca3k6/CWw4at0EcbtGHKJliOW+hSKpN1xiVZfc=" } }, { "type": "step", "result": "=22.5^{\\circ\\:}+\\frac{180^{\\circ\\:}n}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OYxlL+/ci3l7qbr2GAQpO0M8z0oIB8+sCSYYiHIRJRYIM/PMcB7GCA29aob03Bi0J1UGfY00IVmqDhmhWHpTCqORWLXkjysF58uLgjK3bCsjG2ks2HoHn2Q2FV688CriDcfRZrAzcQA/vs00GUDvnmdg6HHLECr/dJQA5cvGc/B6pfF1z6umzUJTJvt+ojYZY4LVEbDWfJhdqtpUA6/OhxHS1SQqIugT5EGbyavpJSLn/7Pc0EBvhuFv6y80xJTAm9GwB1/mEPhWOVtIBAB+Bg==" } }, { "type": "step", "result": "θ=22.5^{\\circ\\:}+\\frac{180^{\\circ\\:}n}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=22.5^{\\circ\\:}+\\frac{180^{\\circ\\:}n}{2}" }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:θ\\le\\:360^{\\circ\\:}$$", "result": "θ=22.5^{\\circ\\:},\\:θ=112.5^{\\circ\\:},\\:θ=202.5^{\\circ\\:},\\:θ=292.5^{\\circ\\:}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\tan(2θ)-1,0^{\\circ }\\le θ\\le 360^{\\circ }" }, "showViewLarger": true } }, "meta": { "showVerify": true } }