3tan^2(θ)+sqrt(3)tan(θ)=0,0<= θ<= 360

{ "query": { "display": "$$3\\tan^{2}\\left(θ\\right)+\\sqrt{3}\\tan\\left(θ\\right)=0,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}$$", "symbolab_question": "EQUATION#3\\tan^{2}(θ)+\\sqrt{3}\\tan(θ)=0,0^{\\circ }\\le θ\\le 360^{\\circ }" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=0,θ=180^{\\circ },θ=360^{\\circ },θ=150^{\\circ },θ=330^{\\circ }", "radians": "θ=0,θ=π,θ=2π,θ=\\frac{5π}{6},θ=\\frac{11π}{6}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$3\\tan^{2}\\left(θ\\right)+\\sqrt{3}\\tan\\left(θ\\right)=0,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}{\\quad:\\quad}θ=0,\\:θ=180^{\\circ\\:},\\:θ=360^{\\circ\\:},\\:θ=150^{\\circ\\:},\\:θ=330^{\\circ\\:}$$", "input": "3\\tan^{2}\\left(θ\\right)+\\sqrt{3}\\tan\\left(θ\\right)=0,\\:0^{\\circ\\:}\\le\\:θ\\le\\:360^{\\circ\\:}", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "3\\tan^{2}\\left(θ\\right)+\\sqrt{3}\\tan\\left(θ\\right)=0", "result": "\\tan\\left(θ\\right)=0,\\:\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Let: $$\\tan\\left(θ\\right)=u$$", "result": "3u^{2}+\\sqrt{3}u=0" }, { "type": "interim", "title": "$$3u^{2}+\\sqrt{3}u=0{\\quad:\\quad}u=0,\\:u=-\\frac{\\sqrt{3}}{3}$$", "input": "3u^{2}+\\sqrt{3}u=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "3u^{2}+\\sqrt{3}u=0", "result": "{u}_{1,\\:2}=\\frac{-\\sqrt{3}\\pm\\:\\sqrt{\\left(\\sqrt{3}\\right)^{2}-4\\cdot\\:3\\cdot\\:0}}{2\\cdot\\:3}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=3,\\:b=\\sqrt{3},\\:c=0$$", "result": "{u}_{1,\\:2}=\\frac{-\\sqrt{3}\\pm\\:\\sqrt{\\left(\\sqrt{3}\\right)^{2}-4\\cdot\\:3\\cdot\\:0}}{2\\cdot\\:3}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{\\left(\\sqrt{3}\\right)^{2}-4\\cdot\\:3\\cdot\\:0}=\\sqrt{3}$$", "input": "\\sqrt{\\left(\\sqrt{3}\\right)^{2}-4\\cdot\\:3\\cdot\\:0}", "result": "{u}_{1,\\:2}=\\frac{-\\sqrt{3}\\pm\\:\\sqrt{3}}{2\\cdot\\:3}", "steps": [ { "type": "interim", "title": "$$\\left(\\sqrt{3}\\right)^{2}=3$$", "input": "\\left(\\sqrt{3}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$", "result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$", "result": "=3^{\\frac{1}{2}\\cdot\\:2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "interim", "title": "$$\\frac{1}{2}\\cdot\\:2=1$$", "input": "\\frac{1}{2}\\cdot\\:2", "result": "=3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XEzlndel0x/QlMW8VTD5UXyRHuGw7+tM5METTDj6vVEMSTe+hEEEr2+K9b3W9JkG8SrqrDW4mFcEK+hPNqZN8jCFrm8vCvLZxZsdY4NLQVuwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$4\\cdot\\:3\\cdot\\:0=0$$", "input": "4\\cdot\\:3\\cdot\\:0", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fVpMbrjjLKU8f2Xgsw8zZiD/swv2EbFFv+X+7iEc3n+jkVi15I8rBefLi4Iyt2wr8D4yaPBYvrqNvcxJbQLVhIRwPCuzRyHq6v39GP1IPwdXeV03GYdXLZ3ZnsTzJuK6" } }, { "type": "step", "result": "=\\sqrt{3-0}" }, { "type": "step", "primary": "Subtract the numbers: $$3-0=3$$", "result": "=\\sqrt{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WXStXFVx75hYDCxA7hRbNIwH0d55OJFfLnIMtXmD8XSTD2Qjcdii8v8hpF3X9+1V3XeO2tIUPH5Q2xrCOU6NXZZAd3iF5cHdgKS1MqMfNWb/P/+v51eTuH2/F4MIu5mqsnA7ORzFfDZOOrgxNtEKIgHfEXGOANZZpLrlmtnSdAGPznlv7iyc0hl+OyWVieJPxDkNFExQYbedEcLN1zmsRg==" } }, { "type": "step", "primary": "Separate the solutions", "result": "{u}_{1}=\\frac{-\\sqrt{3}+\\sqrt{3}}{2\\cdot\\:3},\\:{u}_{2}=\\frac{-\\sqrt{3}-\\sqrt{3}}{2\\cdot\\:3}" }, { "type": "interim", "title": "$$u=\\frac{-\\sqrt{3}+\\sqrt{3}}{2\\cdot\\:3}:{\\quad}0$$", "input": "\\frac{-\\sqrt{3}+\\sqrt{3}}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Add similar elements: $$-\\sqrt{3}+\\sqrt{3}=0$$", "result": "=\\frac{0}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{0}{6}" }, { "type": "step", "primary": "Apply rule $$\\frac{0}{a}=0,\\:a\\ne\\:0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70c9DTe1ubFlIcViNwxWNeevlT6hxQC+Th4ETfpdrq0rBBAGj7i8X1rw2m3/gq2FRA585Wz2Y8ioMtXlAhbC3ebgNL2ycNF0YUgQmiI0KpBZ18c0fw98ackecl02dmEBdhgC0wLaMtvqbwGZDIKE0LFWJeWbliXLz0lre0eyRVKWwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "$$u=\\frac{-\\sqrt{3}-\\sqrt{3}}{2\\cdot\\:3}:{\\quad}-\\frac{\\sqrt{3}}{3}$$", "input": "\\frac{-\\sqrt{3}-\\sqrt{3}}{2\\cdot\\:3}", "steps": [ { "type": "step", "primary": "Add similar elements: $$-\\sqrt{3}-\\sqrt{3}=-2\\sqrt{3}$$", "result": "=\\frac{-2\\sqrt{3}}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{-2\\sqrt{3}}{6}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2\\sqrt{3}}{6}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{\\sqrt{3}}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s70c9DTe1ubFlIcViNwxWNebcdo+cYmRqNPskjj+ABxorBBAGj7i8X1rw2m3/gq2FRA585Wz2Y8ioMtXlAhbC3eQC+BZ/WO81NfjVKT124x6GsxZuKnmRU41UtrFNK/62qR9Duk6WRAyiNU9ojCvKnufQyufhhdJErYImJgcVzaj7MpkJ1YT5ZwLLz78JnRfcweIGQsuMPnBML5aepNx0nIg==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "u=0,\\:u=-\\frac{\\sqrt{3}}{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(θ\\right)$$", "result": "\\tan\\left(θ\\right)=0,\\:\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\tan\\left(θ\\right)=0,\\:0\\le\\:θ\\le\\:360^{\\circ\\:}{\\quad:\\quad}θ=0,\\:θ=180^{\\circ\\:},\\:θ=360^{\\circ\\:}$$", "input": "\\tan\\left(θ\\right)=0,\\:0\\le\\:θ\\le\\:360^{\\circ\\:}", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(θ\\right)=0$$", "result": "θ=0+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": "θ=0+180^{\\circ\\:}n" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$θ=0+180^{\\circ\\:}n:{\\quad}θ=180^{\\circ\\:}n$$", "input": "θ=0+180^{\\circ\\:}n", "steps": [ { "type": "step", "primary": "$$0+180^{\\circ\\:}n=180^{\\circ\\:}n$$", "result": "θ=180^{\\circ\\:}n" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=180^{\\circ\\:}n" }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:θ\\le\\:360^{\\circ\\:}$$", "result": "θ=0,\\:θ=180^{\\circ\\:},\\:θ=360^{\\circ\\:}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3},\\:0\\le\\:θ\\le\\:360^{\\circ\\:}{\\quad:\\quad}θ=150^{\\circ\\:},\\:θ=330^{\\circ\\:}$$", "input": "\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3},\\:0\\le\\:θ\\le\\:360^{\\circ\\:}", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(θ\\right)=-\\frac{\\sqrt{3}}{3}$$", "result": "θ=150^{\\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": "θ=150^{\\circ\\:}+180^{\\circ\\:}n" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "step", "primary": "Solutions for the range $$0\\le\\:θ\\le\\:360^{\\circ\\:}$$", "result": "θ=150^{\\circ\\:},\\:θ=330^{\\circ\\:}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=0,\\:θ=180^{\\circ\\:},\\:θ=360^{\\circ\\:},\\:θ=150^{\\circ\\:},\\:θ=330^{\\circ\\:}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "3\\tan^{2}(θ)+\\sqrt{3}\\tan(θ),0^{\\circ }\\le θ\\le 360^{\\circ }" }, "showViewLarger": true } }, "meta": { "showVerify": true } }