tan(2x)sin(x)-sin(x)-tan(2x)+1=0

{ "query": { "display": "$$\\tan\\left(2x\\right)\\sin\\left(x\\right)-\\sin\\left(x\\right)-\\tan\\left(2x\\right)+1=0$$", "symbolab_question": "EQUATION#\\tan(2x)\\sin(x)-\\sin(x)-\\tan(2x)+1=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{π}{8}+\\frac{πn}{2},x=\\frac{π}{2}+2πn", "degrees": "x=22.5^{\\circ }+90^{\\circ }n,x=90^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan\\left(2x\\right)\\sin\\left(x\\right)-\\sin\\left(x\\right)-\\tan\\left(2x\\right)+1=0{\\quad:\\quad}x=\\frac{π}{8}+\\frac{πn}{2},\\:x=\\frac{π}{2}+2πn$$", "input": "\\tan\\left(2x\\right)\\sin\\left(x\\right)-\\sin\\left(x\\right)-\\tan\\left(2x\\right)+1=0", "steps": [ { "type": "interim", "title": "Factor $$\\tan\\left(2x\\right)\\sin\\left(x\\right)-\\sin\\left(x\\right)-\\tan\\left(2x\\right)+1:{\\quad}\\left(\\tan\\left(2x\\right)-1\\right)\\left(\\sin\\left(x\\right)-1\\right)$$", "input": "\\tan\\left(2x\\right)\\sin\\left(x\\right)-\\sin\\left(x\\right)-\\tan\\left(2x\\right)+1", "steps": [ { "type": "step", "result": "=\\left(\\sin\\left(x\\right)\\tan\\left(2x\\right)-\\sin\\left(x\\right)\\right)+\\left(-\\tan\\left(2x\\right)+1\\right)" }, { "type": "interim", "title": "Factor out $$-1\\:$$from $$-\\tan\\left(2x\\right)+1:\\quad\\:-\\left(\\tan\\left(2x\\right)-1\\right)$$", "input": "-\\tan\\left(2x\\right)+1", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(\\tan\\left(2x\\right)-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XCHcf9tERCRpuDrcQoZd4npgCW63KSKi1YfMrlRapV4H9A8JFyuzjx7kca+eJh9pxjKoMzc0d1qWN5qd2LftJhO15vwW5AkXXbjpCHLC7iCjeh7+jKEzLb7VNCEMF3Z/y2htv8Bp5enTcNx/Zte17E4Z1JW8gRH1yQlpQrZAiwGPjZ23YV91Ch+hZaTutXD1M1z5WTj26yU6IwaAs+4KOw==" } }, { "type": "interim", "title": "Factor out $$\\sin\\left(x\\right)\\:$$from $$\\sin\\left(x\\right)\\tan\\left(2x\\right)-\\sin\\left(x\\right):\\quad\\:\\sin\\left(x\\right)\\left(\\tan\\left(2x\\right)-1\\right)$$", "input": "\\sin\\left(x\\right)\\tan\\left(2x\\right)-\\sin\\left(x\\right)", "steps": [ { "type": "step", "primary": "Factor out common term $$\\sin\\left(x\\right)$$", "result": "=\\sin\\left(x\\right)\\left(\\tan\\left(2x\\right)-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Awm8OHaUUE5X73KqyTFjFEa9SOr59lmdJH/sooCdsXg98R45bR+LZcawua215z9qA4IthQdsg7ucXFDvKEaf0CAc/hipaSHk44dpKtD+XxFc7wREDGZ57MDi7uEY2A+q8MlYl8MaKpZvU1tAr7PfmYXho5UtoG65UpL2Woz33DByCLLZFXzLfhcqOiHyUnGpn1DF/A9C5KctEfzRMdDquQy1acS6hLW/Z8k3fywnMBtCwbh/9z8xNdWNmHBsXa5d" } }, { "type": "step", "result": "=-\\left(\\tan\\left(2x\\right)-1\\right)+\\sin\\left(x\\right)\\left(\\tan\\left(2x\\right)-1\\right)" }, { "type": "step", "primary": "Factor out common term $$\\tan\\left(2x\\right)-1$$", "result": "=\\left(\\tan\\left(2x\\right)-1\\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": "\\left(\\tan\\left(2x\\right)-1\\right)\\left(\\sin\\left(x\\right)-1\\right)=0" }, { "type": "step", "primary": "Solving each part separately", "result": "\\tan\\left(2x\\right)-1=0\\lor\\:\\sin\\left(x\\right)-1=0" }, { "type": "interim", "title": "$$\\tan\\left(2x\\right)-1=0{\\quad:\\quad}x=\\frac{π}{8}+\\frac{πn}{2}$$", "input": "\\tan\\left(2x\\right)-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "\\tan\\left(2x\\right)-1=0", "result": "\\tan\\left(2x\\right)=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "\\tan\\left(2x\\right)-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "\\tan\\left(2x\\right)=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\tan\\left(2x\\right)=1$$", "result": "2x=\\frac{π}{4}+πn", "steps": [ { "type": "step", "primary": "$$\\tan\\left(x\\right)$$ periodicity table with $$πn$$ cycle:<br/>$$\\begin{array}{|c|c|}\\hline x&\\tan(x)\\\\\\hline 0&0\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{3}\\\\\\hline \\frac{π}{4}&1\\\\\\hline \\frac{π}{3}&\\sqrt{3}\\\\\\hline \\frac{π}{2}&\\pm\\infty\\\\\\hline \\frac{2π}{3}&-\\sqrt{3}\\\\\\hline \\frac{3π}{4}&-1\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{3}\\\\\\hline &\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "2x=\\frac{π}{4}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{π}{4}+πn:{\\quad}x=\\frac{π}{8}+\\frac{πn}{2}$$", "input": "2x=\\frac{π}{4}+πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{π}{4}+πn", "result": "x=\\frac{π}{8}+\\frac{πn}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{π}{4}}{2}+\\frac{πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{π}{4}}{2}+\\frac{πn}{2}", "result": "x=\\frac{π}{8}+\\frac{πn}{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{2x}{2}:{\\quad}x$$", "input": "\\frac{2x}{2}", "steps": [ { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wyZUoCmW9j1Yiq04nFm+Ei061ljBSPJeENOw2efoSWt2cZ6LAXMcKee6PSyjLEFrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6rogJeWDMKCUBljFmQOLy/H" } }, { "type": "interim", "title": "Simplify $$\\frac{\\frac{π}{4}}{2}+\\frac{πn}{2}:{\\quad}\\frac{π}{8}+\\frac{πn}{2}$$", "input": "\\frac{\\frac{π}{4}}{2}+\\frac{πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{π}{4}}{2}=\\frac{π}{8}$$", "input": "\\frac{\\frac{π}{4}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{π}{4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=\\frac{π}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajfBUDCK2p4efPsnQUA32SOjNGoPE9TME3q+OPmgkv2RQV7wanvTgjKqnID/UMq66G76DMDH258ED6TCmtXj8Vmq+/XUfFY44oQy7yrU6sldleyoAWyEiMC09UXmm0ysSzygo1WVqYqDorj9mXvCvqes=" } }, { "type": "step", "result": "=\\frac{π}{8}+\\frac{πn}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajXnqeQkVqfwvjiIs6Ts5NPxWUBZ6dJyds/Ck/nc7uAfbA585Wz2Y8ioMtXlAhbC3eU6oQ6YxiIai+QHVGEfQ+0DM7EkRtc+FHgZlFRX75Xc5Ec7ShOedm97LMngC0LVkYx4pgUWEah0lniZLlD4X0wtEEedfW/3lEhzme5c9xQJXb2NjPCr8WnDQrhnECKgGpDKat+HVL7bXLjEPppCiZm0=" } }, { "type": "step", "result": "x=\\frac{π}{8}+\\frac{πn}{2}" } ], "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}{8}+\\frac{πn}{2}" } ], "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}{8}+\\frac{πn}{2},\\:x=\\frac{π}{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": "\\tan(2x)\\sin(x)-\\sin(x)-\\tan(2x)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }