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

{ "query": { "display": "$$2\\sin\\left(x\\right)-\\cos\\left(x\\right)\\sin\\left(x\\right)=0$$", "symbolab_question": "EQUATION#2\\sin(x)-\\cos(x)\\sin(x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$2\\sin\\left(x\\right)-\\cos\\left(x\\right)\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "2\\sin\\left(x\\right)-\\cos\\left(x\\right)\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Factor $$2\\sin\\left(x\\right)-\\cos\\left(x\\right)\\sin\\left(x\\right):{\\quad}-\\sin\\left(x\\right)\\left(\\cos\\left(x\\right)-2\\right)$$", "input": "2\\sin\\left(x\\right)-\\cos\\left(x\\right)\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$-\\sin\\left(x\\right)$$", "result": "=-\\sin\\left(x\\right)\\left(-2+\\cos\\left(x\\right)\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "-\\sin\\left(x\\right)\\left(\\cos\\left(x\\right)-2\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\sin\\left(x\\right)=0\\lor\\:\\cos\\left(x\\right)-2=0" }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=0$$", "result": "x=0+2πn,\\:x=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=0+2πn,\\:x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\cos\\left(x\\right)-2=0{\\quad:\\quad}$$No Solution", "input": "\\cos\\left(x\\right)-2=0", "steps": [ { "type": "interim", "title": "Move $$2\\:$$to the right side", "input": "\\cos\\left(x\\right)-2=0", "result": "\\cos\\left(x\\right)=2", "steps": [ { "type": "step", "primary": "Add $$2$$ to both sides", "result": "\\cos\\left(x\\right)-2+2=0+2" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(x\\right)=2" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "step", "primary": "$$-1\\le\\cos\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "2\\sin(x)-\\cos(x)\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }