sin(x)= 4/(cos(x))

{ "query": { "display": "$$\\sin\\left(x\\right)=\\frac{4}{\\cos\\left(x\\right)}$$", "symbolab_question": "EQUATION#\\sin(x)=\\frac{4}{\\cos(x)}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{4}{\\cos\\left(x\\right)}{\\quad:\\quad}$$No Solution for $$x\\in\\mathbb{R}$$", "input": "\\sin\\left(x\\right)=\\frac{4}{\\cos\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Subtract $$\\frac{4}{\\cos\\left(x\\right)}$$ from both sides", "result": "\\sin\\left(x\\right)-\\frac{4}{\\cos\\left(x\\right)}=0" }, { "type": "interim", "title": "Simplify $$\\sin\\left(x\\right)-\\frac{4}{\\cos\\left(x\\right)}:{\\quad}\\frac{\\sin\\left(x\\right)\\cos\\left(x\\right)-4}{\\cos\\left(x\\right)}$$", "input": "\\sin\\left(x\\right)-\\frac{4}{\\cos\\left(x\\right)}", "steps": [ { "type": "step", "primary": "Convert element to fraction: $$\\sin\\left(x\\right)=\\frac{\\sin\\left(x\\right)\\cos\\left(x\\right)}{\\cos\\left(x\\right)}$$", "result": "=\\frac{\\sin\\left(x\\right)\\cos\\left(x\\right)}{\\cos\\left(x\\right)}-\\frac{4}{\\cos\\left(x\\right)}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{\\sin\\left(x\\right)\\cos\\left(x\\right)-4}{\\cos\\left(x\\right)}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "\\frac{\\sin\\left(x\\right)\\cos\\left(x\\right)-4}{\\cos\\left(x\\right)}=0" }, { "type": "step", "primary": "$$\\frac{f\\left(x\\right)}{g\\left(x\\right)}=0{\\quad\\Rightarrow\\quad}f\\left(x\\right)=0$$", "result": "\\sin\\left(x\\right)\\cos\\left(x\\right)-4=0" }, { "type": "interim", "title": "Rewrite using trig identities", "input": "\\sin\\left(x\\right)\\cos\\left(x\\right)-4", "result": "-4+\\frac{\\sin\\left(2x\\right)}{2}=0", "steps": [ { "type": "step", "primary": "Use the Double Angle identity: $$2\\sin\\left(x\\right)\\cos\\left(x\\right)=\\sin\\left(2x\\right)$$", "secondary": [ "$$\\sin\\left(x\\right)\\cos\\left(x\\right)=\\frac{\\sin\\left(2x\\right)}{2}$$" ], "result": "=-4+\\frac{\\sin\\left(2x\\right)}{2}" } ], "meta": { "interimType": "Trig Rewrite Using Trig identities 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFEmv8hbufrMaqLvi1Nsm6seyuRcnFgnh8AijhYNjhqyRbyZYyX6EGMAcAMgjOADqPLajIxP2OrjkBCECQd8a6w5NGl0QxAbgiQb78/w1L9Vff8Smu/+n4LISRk5JEw/5bK5yACf120kzY7jSWvmAiTR2WVG1dxuhtwcvc5+As2ToKO+d89V0ksUsI4xf7WSo4wJYYPwif7On5mZsZiNIHrw=" } }, { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "-4+\\frac{\\sin\\left(2x\\right)}{2}=0", "result": "\\frac{\\sin\\left(2x\\right)}{2}=4", "steps": [ { "type": "step", "primary": "Add $$4$$ to both sides", "result": "-4+\\frac{\\sin\\left(2x\\right)}{2}+4=0+4" }, { "type": "step", "primary": "Simplify", "result": "\\frac{\\sin\\left(2x\\right)}{2}=4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{\\sin\\left(2x\\right)}{2}=4", "result": "\\sin\\left(2x\\right)=8", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{2\\sin\\left(2x\\right)}{2}=4\\cdot\\:2" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(2x\\right)=8" } ], "meta": { "interimType": "Multiply Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution\\:for}\\:x\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(x)-\\frac{4}{\\cos(x)}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }