solvefor x,sin(xθ)= 1/2

{ "query": { "display": "solve for $$x,\\:\\sin\\left(xθ\\right)=\\frac{1}{2}$$", "symbolab_question": "SOLVE_FOR#solvefor x,\\sin(xθ)=\\frac{1}{2}" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{6θ}+\\frac{2πn}{θ},x=\\frac{5π}{6θ}+\\frac{2πn}{θ};\\quad θ\\ne 0", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$solve\\:for\\:x,\\:\\sin\\left(xθ\\right)=\\frac{1}{2}{\\quad:\\quad}x=\\frac{π}{6θ}+\\frac{2πn}{θ},\\:x=\\frac{5π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0$$", "input": "\\sin\\left(xθ\\right)=\\frac{1}{2}", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(xθ\\right)=\\frac{1}{2}$$", "result": "xθ=\\frac{π}{6}+2πn,\\:xθ=\\frac{5π}{6}+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": "xθ=\\frac{π}{6}+2πn,\\:xθ=\\frac{5π}{6}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$xθ=\\frac{π}{6}+2πn:{\\quad}x=\\frac{π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0$$", "input": "xθ=\\frac{π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$θ;\\quad\\:θ\\ne\\:0$$", "input": "xθ=\\frac{π}{6}+2πn", "result": "x=\\frac{π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0", "steps": [ { "type": "step", "primary": "Divide both sides by $$θ;\\quad\\:θ\\ne\\:0$$", "result": "\\frac{xθ}{θ}=\\frac{\\frac{π}{6}}{θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0" }, { "type": "interim", "title": "Simplify", "input": "\\frac{xθ}{θ}=\\frac{\\frac{π}{6}}{θ}+\\frac{2πn}{θ}", "result": "x=\\frac{π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{xθ}{θ}:{\\quad}x$$", "input": "\\frac{xθ}{θ}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$θ$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/fjMhIaPVCWP/icuYbNUmXyRHuGw7+tM5METTDj6vVEed1oZgJr3Rrt+25B4RF18ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz3AXOMi6S8kb1ZHQGJZHZD6ialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{6}}{θ}+\\frac{2πn}{θ}:{\\quad}\\frac{π}{6θ}+\\frac{2πn}{θ}$$", "input": "\\frac{\\frac{π}{6}}{θ}+\\frac{2πn}{θ}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{6θ}+\\frac{2πn}{θ}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajTw8zVWgUh6BPCRhFM4sQDswqOl1Wqr7iu+XsZ06B+KHICf2WQN9mSJxQaQ7cQX4il2BgpogRffZXrJl2fYiHyc+dr0GjudUTYm9vAbCfoR6B/ap7gHA+DkXY8jhHrQXW9bA+zX4bD3u3gx65o2NJhPA8EHDdvIHlE+2lLLpgu3aGq8gGCBW8+lRkRpTpkzS/TTfsuvIlslexra8HoCd/6zHo4oOIrTJ/7hiOyX/PIro" } }, { "type": "step", "result": "x=\\frac{π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$xθ=\\frac{5π}{6}+2πn:{\\quad}x=\\frac{5π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0$$", "input": "xθ=\\frac{5π}{6}+2πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$θ;\\quad\\:θ\\ne\\:0$$", "input": "xθ=\\frac{5π}{6}+2πn", "result": "x=\\frac{5π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0", "steps": [ { "type": "step", "primary": "Divide both sides by $$θ;\\quad\\:θ\\ne\\:0$$", "result": "\\frac{xθ}{θ}=\\frac{\\frac{5π}{6}}{θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0" }, { "type": "interim", "title": "Simplify", "input": "\\frac{xθ}{θ}=\\frac{\\frac{5π}{6}}{θ}+\\frac{2πn}{θ}", "result": "x=\\frac{5π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{xθ}{θ}:{\\quad}x$$", "input": "\\frac{xθ}{θ}", "steps": [ { "type": "step", "primary": "Cancel the common factor: $$θ$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/fjMhIaPVCWP/icuYbNUmXyRHuGw7+tM5METTDj6vVEed1oZgJr3Rrt+25B4RF18ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz3AXOMi6S8kb1ZHQGJZHZD6ialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{5π}{6}}{θ}+\\frac{2πn}{θ}:{\\quad}\\frac{5π}{6θ}+\\frac{2πn}{θ}$$", "input": "\\frac{\\frac{5π}{6}}{θ}+\\frac{2πn}{θ}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{5π}{6θ}+\\frac{2πn}{θ}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajZecAISRhHNtihkpQDVIN03NOXBIx3x2FSRwIWJr3Z8K3oZCq59Hq2va8/E5S/sf7wVWfb8eY836P59a0g0AIKZMzX4V1VwSHFRMXh1b8snfQiyp3LE30X8fD7I9aGe7Y4EFMST8lDZxn1Yq5HMKVTtAb51BceE+a5ppiVMbLOTZUA2Y3m9nPXY03t5CrS6u9mzk/QAcaAibpmQa1Lz33bYOPTySRv5OG5ug8PqzCbHu" } }, { "type": "step", "result": "x=\\frac{5π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{π}{6θ}+\\frac{2πn}{θ},\\:x=\\frac{5π}{6θ}+\\frac{2πn}{θ};\\quad\\:θ\\ne\\:0" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\sin(xθ)-\\frac{1}{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }