-2cot(θ)=cot^2(θ)+1

{ "query": { "display": "$$-2\\cot\\left(θ\\right)=\\cot^{2}\\left(θ\\right)+1$$", "symbolab_question": "EQUATION#-2\\cot(θ)=\\cot^{2}(θ)+1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{3π}{4}+πn", "degrees": "θ=135^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-2\\cot\\left(θ\\right)=\\cot^{2}\\left(θ\\right)+1{\\quad:\\quad}θ=\\frac{3π}{4}+πn$$", "input": "-2\\cot\\left(θ\\right)=\\cot^{2}\\left(θ\\right)+1", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "-2\\cot\\left(θ\\right)=\\cot^{2}\\left(θ\\right)+1", "result": "\\cot\\left(θ\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cot\\left(θ\\right)=u$$", "result": "-2u=u^{2}+1" }, { "type": "interim", "title": "$$-2u=u^{2}+1{\\quad:\\quad}u=-1$$", "input": "-2u=u^{2}+1", "steps": [ { "type": "step", "primary": "Switch sides", "result": "u^{2}+1=-2u" }, { "type": "interim", "title": "Move $$2u\\:$$to the left side", "input": "u^{2}+1=-2u", "result": "u^{2}+1+2u=0", "steps": [ { "type": "step", "primary": "Add $$2u$$ to both sides", "result": "u^{2}+1+2u=-2u+2u" }, { "type": "step", "primary": "Simplify", "result": "u^{2}+1+2u=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "Write in the standard form $$ax^{2}+bx+c=0$$", "result": "u^{2}+2u+1=0" }, { "type": "interim", "title": "Solve with the quadratic formula", "input": "u^{2}+2u+1=0", "result": "{u}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:1}}{2\\cdot\\:1}", "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 $${\\quad}a=1,\\:b=2,\\:c=1$$", "result": "{u}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:1}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$2^{2}-4\\cdot\\:1\\cdot\\:1=0$$", "input": "2^{2}-4\\cdot\\:1\\cdot\\:1", "result": "{u}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{0}}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:1=4$$", "result": "=2^{2}-4" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=4-4" }, { "type": "step", "primary": "Subtract the numbers: $$4-4=0$$", "result": "=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7GEEX0EiNBYzwhkunUPgFxYcFPqa2dwwYBvRbhQUK9o4DnzlbPZjyKgy1eUCFsLd5uA0vbJw0XRhSBCaIjQqkFmVBHPdZOPO8WPbmsZqQtxm6v4V6Vpg/OWLsuIqa/9fQsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "step", "result": "u=\\frac{-2}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$\\frac{-2}{2\\cdot\\:1}=-1$$", "input": "\\frac{-2}{2\\cdot\\:1}", "result": "u=-1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-2}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pQ0w2CFUmP3V08m8YszlLWgelhrK8M2TY4B4ebvtusYJQJZuTAY5js+oqjdT8kslMsz7gcINLRJHih26h/L+6fTmpwXN29XlGx6TKevDbs/ZlQvGb17FQnE0KcvUlNOsJLd1ohke2Wgml78++2zI0g==" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cot\\left(θ\\right)$$", "result": "\\cot\\left(θ\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cot\\left(θ\\right)=-1{\\quad:\\quad}θ=\\frac{3π}{4}+πn$$", "input": "\\cot\\left(θ\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cot\\left(θ\\right)=-1$$", "result": "θ=\\frac{3π}{4}+πn", "steps": [ { "type": "step", "primary": "$$\\cot\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\cot(x)\\\\\\hline 0&\\mp\\infty\\\\\\hline \\frac{π}{6}&\\sqrt{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\sqrt{3}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "θ=\\frac{3π}{4}+πn" } ], "meta": { "interimType": "Trig General Solutions cot 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "θ=\\frac{3π}{4}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "-2\\cot(θ)-\\cot^{2}(θ)-1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }