-cos(a)-11=2cos(a)-9

{ "query": { "display": "$$-\\cos\\left(a\\right)-11=2\\cos\\left(a\\right)-9$$", "symbolab_question": "EQUATION#-\\cos(a)-11=2\\cos(a)-9" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "a=2.30052…+2πn,a=-2.30052…+2πn", "degrees": "a=131.81031…^{\\circ }+360^{\\circ }n,a=-131.81031…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$-\\cos\\left(a\\right)-11=2\\cos\\left(a\\right)-9{\\quad:\\quad}a=2.30052…+2πn,\\:a=-2.30052…+2πn$$", "input": "-\\cos\\left(a\\right)-11=2\\cos\\left(a\\right)-9", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "-\\cos\\left(a\\right)-11=2\\cos\\left(a\\right)-9", "result": "\\cos\\left(a\\right)=-\\frac{2}{3}", "steps": [ { "type": "step", "primary": "Let: $$\\cos\\left(a\\right)=u$$", "result": "-u-11=2u-9" }, { "type": "interim", "title": "$$-u-11=2u-9{\\quad:\\quad}u=-\\frac{2}{3}$$", "input": "-u-11=2u-9", "steps": [ { "type": "interim", "title": "Move $$11\\:$$to the right side", "input": "-u-11=2u-9", "result": "-u=2u+2", "steps": [ { "type": "step", "primary": "Add $$11$$ to both sides", "result": "-u-11+11=2u-9+11" }, { "type": "step", "primary": "Simplify", "result": "-u=2u+2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Move $$2u\\:$$to the left side", "input": "-u=2u+2", "result": "-3u=2", "steps": [ { "type": "step", "primary": "Subtract $$2u$$ from both sides", "result": "-u-2u=2u+2-2u" }, { "type": "step", "primary": "Simplify", "result": "-3u=2" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-3$$", "input": "-3u=2", "result": "u=-\\frac{2}{3}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-3$$", "result": "\\frac{-3u}{-3}=\\frac{2}{-3}" }, { "type": "step", "primary": "Simplify", "result": "u=-\\frac{2}{3}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\cos\\left(a\\right)$$", "result": "\\cos\\left(a\\right)=-\\frac{2}{3}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\cos\\left(a\\right)=-\\frac{2}{3}{\\quad:\\quad}a=\\arccos\\left(-\\frac{2}{3}\\right)+2πn,\\:a=-\\arccos\\left(-\\frac{2}{3}\\right)+2πn$$", "input": "\\cos\\left(a\\right)=-\\frac{2}{3}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(a\\right)=-\\frac{2}{3}", "result": "a=\\arccos\\left(-\\frac{2}{3}\\right)+2πn,\\:a=-\\arccos\\left(-\\frac{2}{3}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(a\\right)=-\\frac{2}{3}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "a=\\arccos\\left(-\\frac{2}{3}\\right)+2πn,\\:a=-\\arccos\\left(-\\frac{2}{3}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "a=\\arccos\\left(-\\frac{2}{3}\\right)+2πn,\\:a=-\\arccos\\left(-\\frac{2}{3}\\right)+2πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "a=2.30052…+2πn,\\:a=-2.30052…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "a", "plotRequest": "-\\cos(a)-11-2\\cos(a)+9" }, "showViewLarger": true } }, "meta": { "showVerify": true } }