5cos(c)+sqrt(23)=0

{ "query": { "display": "$$5\\cos\\left(c\\right)+\\sqrt{23}=0$$", "symbolab_question": "EQUATION#5\\cos(c)+\\sqrt{23}=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "c=2.85483…+2πn,c=-2.85483…+2πn", "degrees": "c=163.57005…^{\\circ }+360^{\\circ }n,c=-163.57005…^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$5\\cos\\left(c\\right)+\\sqrt{23}=0{\\quad:\\quad}c=2.85483…+2πn,\\:c=-2.85483…+2πn$$", "input": "5\\cos\\left(c\\right)+\\sqrt{23}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{23}\\:$$to the right side", "input": "5\\cos\\left(c\\right)+\\sqrt{23}=0", "result": "5\\cos\\left(c\\right)=-\\sqrt{23}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{23}$$ from both sides", "result": "5\\cos\\left(c\\right)+\\sqrt{23}-\\sqrt{23}=0-\\sqrt{23}" }, { "type": "step", "primary": "Simplify", "result": "5\\cos\\left(c\\right)=-\\sqrt{23}" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5\\cos\\left(c\\right)=-\\sqrt{23}", "result": "\\cos\\left(c\\right)=-\\frac{\\sqrt{23}}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5\\cos\\left(c\\right)}{5}=\\frac{-\\sqrt{23}}{5}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(c\\right)=-\\frac{\\sqrt{23}}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\cos\\left(c\\right)=-\\frac{\\sqrt{23}}{5}", "result": "c=\\arccos\\left(-\\frac{\\sqrt{23}}{5}\\right)+2πn,\\:c=-\\arccos\\left(-\\frac{\\sqrt{23}}{5}\\right)+2πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\cos\\left(c\\right)=-\\frac{\\sqrt{23}}{5}$$", "secondary": [ "$$\\cos\\left(x\\right)=-a\\quad\\Rightarrow\\quad\\:x=\\arccos\\left(-a\\right)+2πn,\\:\\quad\\:x=-\\arccos\\left(-a\\right)+2πn$$" ], "result": "c=\\arccos\\left(-\\frac{\\sqrt{23}}{5}\\right)+2πn,\\:c=-\\arccos\\left(-\\frac{\\sqrt{23}}{5}\\right)+2πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Show solutions in decimal form", "result": "c=2.85483…+2πn,\\:c=-2.85483…+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "c", "plotRequest": "5\\cos(c)+\\sqrt{23}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }