cos^2(x)+1=-2cos(x)

{ "query": { "display": "$$\\cos^{2}\\left(x\\right)+1=-2\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#\\cos^{2}(x)+1=-2\\cos(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=π+2πn", "degrees": "x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos^{2}\\left(x\\right)+1=-2\\cos\\left(x\\right){\\quad:\\quad}x=π+2πn$$", "input": "\\cos^{2}\\left(x\\right)+1=-2\\cos\\left(x\\right)", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\cos^{2}\\left(x\\right)+1=-2\\cos\\left(x\\right)", "result": "\\cos\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(x\\right)=u$$", "result": "u^{2}+1=-2u" }, { "type": "interim", "title": "$$u^{2}+1=-2u{\\quad:\\quad}u=-1$$", "input": "u^{2}+1=-2u", "steps": [ { "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 $$u=\\cos\\left(x\\right)$$", "result": "\\cos\\left(x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=-1{\\quad:\\quad}x=π+2πn$$", "input": "\\cos\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=-1$$", "result": "x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\cos^{2}(x)+1+2\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }