cos(2θ)=-28/53

{ "query": { "display": "$$\\cos\\left(2θ\\right)=-\\frac{28}{53}$$", "symbolab_question": "EQUATION#\\cos(2θ)=-\\frac{28}{53}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "θ=\\frac{2.12739…}{2}+πn,θ=-\\frac{2.12739…}{2}+πn", "degrees": "θ=60.94539…^{\\circ }+180^{\\circ }n,θ=-60.94539…^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\cos\\left(2θ\\right)=-\\frac{28}{53}{\\quad:\\quad}θ=\\frac{2.12739…}{2}+πn,\\:θ=-\\frac{2.12739…}{2}+πn$$", "input": "\\cos\\left(2θ\\right)=-\\frac{28}{53}", "steps": [ { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(2θ\\right)=-\\frac{28}{53}", "result": "2θ=\\arccos\\left(-\\frac{28}{53}\\right)+2πn,\\:2θ=-\\arccos\\left(-\\frac{28}{53}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(2θ\\right)=-\\frac{28}{53}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "2θ=\\arccos\\left(-\\frac{28}{53}\\right)+2πn,\\:2θ=-\\arccos\\left(-\\frac{28}{53}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "interim", "title": "Solve $$2θ=\\arccos\\left(-\\frac{28}{53}\\right)+2πn:{\\quad}θ=\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn$$", "input": "2θ=\\arccos\\left(-\\frac{28}{53}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2θ=\\arccos\\left(-\\frac{28}{53}\\right)+2πn", "result": "θ=\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2θ}{2}=\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "θ=\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$2θ=-\\arccos\\left(-\\frac{28}{53}\\right)+2πn:{\\quad}θ=-\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn$$", "input": "2θ=-\\arccos\\left(-\\frac{28}{53}\\right)+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2θ=-\\arccos\\left(-\\frac{28}{53}\\right)+2πn", "result": "θ=-\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2θ}{2}=-\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+\\frac{2πn}{2}" }, { "type": "step", "primary": "Simplify", "result": "θ=-\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "θ=\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn,\\:θ=-\\frac{\\arccos\\left(-\\frac{28}{53}\\right)}{2}+πn" }, { "type": "step", "primary": "Show solutions in decimal form", "result": "θ=\\frac{2.12739…}{2}+πn,\\:θ=-\\frac{2.12739…}{2}+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "\\cos(2θ)+\\frac{28}{53}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }