2cos(x)-5sin(x)=0

{ "query": { "display": "$$2\\cos\\left(x\\right)-5\\sin\\left(x\\right)=0$$", "symbolab_question": "EQUATION#2\\cos(x)-5\\sin(x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=0.38050…+πn", "degrees": "x=21.80140…^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\cos\\left(x\\right)-5\\sin\\left(x\\right)=0{\\quad:\\quad}x=0.38050…+πn$$", "input": "2\\cos\\left(x\\right)-5\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Rewrite using trig identities", "input": "2\\cos\\left(x\\right)-5\\sin\\left(x\\right)=0", "result": "2-5\\tan\\left(x\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$\\cos\\left(x\\right),\\:\\cos\\left(x\\right)\\neq0$$", "result": "\\frac{2\\cos\\left(x\\right)-5\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\frac{0}{\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Simplify", "result": "2-\\frac{5\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=0" }, { "type": "step", "primary": "Use the basic trigonometric identity: $$\\frac{\\sin\\left(x\\right)}{\\cos\\left(x\\right)}=\\tan\\left(x\\right)$$", "result": "2-5\\tan\\left(x\\right)=0" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s791AXYceO9Pc4CRAMY99YBeCpSM0lE3MiFWsxPloCjvg52FciCV6Q/ZuTzBHIPdDy6bFjjx5IuxvIy7/nJMTFRsLgcgrrcmkH6sZs7VXsfRC18U/sbftA4WJBD8q7d/ezQ7ycjPUGJf363iUwQP+pSvC30sSftAIFS6Qkpy19IkqTZVB2o8ayZhNpjJ6yWUdNFzgpezzGAk2IuQnCvRv/k4mpXFf3SOUx+H18qfp3MLg=" } }, { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "2-5\\tan\\left(x\\right)=0", "result": "-5\\tan\\left(x\\right)=-2", "steps": [ { "type": "step", "primary": "Subtract $$2$$ from both sides", "result": "2-5\\tan\\left(x\\right)-2=0-2" }, { "type": "step", "primary": "Simplify", "result": "-5\\tan\\left(x\\right)=-2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-5$$", "input": "-5\\tan\\left(x\\right)=-2", "result": "\\tan\\left(x\\right)=\\frac{2}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$-5$$", "result": "\\frac{-5\\tan\\left(x\\right)}{-5}=\\frac{-2}{-5}" }, { "type": "step", "primary": "Simplify", "result": "\\tan\\left(x\\right)=\\frac{2}{5}" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Apply trig inverse properties", "input": "\\tan\\left(x\\right)=\\frac{2}{5}", "result": "x=\\arctan\\left(\\frac{2}{5}\\right)+πn", "steps": [ { "type": "step", "primary": "General solutions for $$\\tan\\left(x\\right)=\\frac{2}{5}$$", "secondary": [ "$$\\tan\\left(x\\right)=a\\quad\\Rightarrow\\quad\\:x=\\arctan\\left(a\\right)+πn$$" ], "result": "x=\\arctan\\left(\\frac{2}{5}\\right)+πn" } ], "meta": { "interimType": "Trig Apply Inverse Props 0Eq" } }, { "type": "step", "primary": "Show solutions in decimal form", "result": "x=0.38050…+πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\cos(x)-5\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }