sin(x)*cos(x)=cos(x)

{ "query": { "display": "$$\\sin\\left(x\\right)\\cdot\\:\\cos\\left(x\\right)=\\cos\\left(x\\right)$$", "symbolab_question": "EQUATION#\\sin(x)\\cdot \\cos(x)=\\cos(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{2}+2πn,x=\\frac{3π}{2}+2πn", "degrees": "x=90^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin\\left(x\\right)\\cos\\left(x\\right)=\\cos\\left(x\\right){\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(x\\right)\\cos\\left(x\\right)=\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Subtract $$\\cos\\left(x\\right)$$ from both sides", "result": "\\sin\\left(x\\right)\\cos\\left(x\\right)-\\cos\\left(x\\right)=0" }, { "type": "interim", "title": "Factor $$\\sin\\left(x\\right)\\cos\\left(x\\right)-\\cos\\left(x\\right):{\\quad}\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-1\\right)$$", "input": "\\sin\\left(x\\right)\\cos\\left(x\\right)-\\cos\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$\\cos\\left(x\\right)$$", "result": "=\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "\\cos\\left(x\\right)\\left(\\sin\\left(x\\right)-1\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\cos\\left(x\\right)=0\\lor\\:\\sin\\left(x\\right)-1=0" }, { "type": "interim", "title": "$$\\cos\\left(x\\right)=0{\\quad:\\quad}x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\cos\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\cos\\left(x\\right)=0$$", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)-1=0{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin\\left(x\\right)-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\sin\\left(x\\right)-1=0", "result": "\\sin\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\sin\\left(x\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(x\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=1$$", "result": "x=\\frac{π}{2}+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=\\frac{π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn" } ], "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)\\cdot \\cos(x)-\\cos(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }