tan^3(2x)-tan^2(2x)-tan(2x)+1=0

{ "query": { "display": "$$\\tan^{3}\\left(2x\\right)-\\tan^{2}\\left(2x\\right)-\\tan\\left(2x\\right)+1=0$$", "symbolab_question": "EQUATION#\\tan^{3}(2x)-\\tan^{2}(2x)-\\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{3π}{8}+\\frac{πn}{2}", "degrees": "x=22.5^{\\circ }+90^{\\circ }n,x=67.5^{\\circ }+90^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan^{3}\\left(2x\\right)-\\tan^{2}\\left(2x\\right)-\\tan\\left(2x\\right)+1=0{\\quad:\\quad}x=\\frac{π}{8}+\\frac{πn}{2},\\:x=\\frac{3π}{8}+\\frac{πn}{2}$$", "input": "\\tan^{3}\\left(2x\\right)-\\tan^{2}\\left(2x\\right)-\\tan\\left(2x\\right)+1=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\tan^{3}\\left(2x\\right)-\\tan^{2}\\left(2x\\right)-\\tan\\left(2x\\right)+1=0", "result": "\\tan\\left(2x\\right)=1,\\:\\tan\\left(2x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\tan\\left(2x\\right)=u$$", "result": "u^{3}-u^{2}-u+1=0" }, { "type": "interim", "title": "$$u^{3}-u^{2}-u+1=0{\\quad:\\quad}u=1,\\:u=-1$$", "input": "u^{3}-u^{2}-u+1=0", "steps": [ { "type": "interim", "title": "Factor $$u^{3}-u^{2}-u+1:{\\quad}\\left(u-1\\right)^{2}\\left(u+1\\right)$$", "input": "u^{3}-u^{2}-u+1", "steps": [ { "type": "step", "result": "=\\left(u^{3}-u^{2}\\right)+\\left(-u+1\\right)" }, { "type": "interim", "title": "Factor out $$-1\\:$$from $$-u+1:\\quad\\:-\\left(u-1\\right)$$", "input": "-u+1", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(u-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7793KOKq2opFbYEfUBMR99sevWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G1J1sKwFl1+Ulr5AICqN7vyKN6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17XsThnUlbyBEfXJCWlCtkCLARH2IYAsS/z5QOTIQpgtYog=" } }, { "type": "interim", "title": "Factor out $$u^{2}\\:$$from $$u^{3}-u^{2}:\\quad\\:u^{2}\\left(u-1\\right)$$", "input": "u^{3}-u^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{3}=uu^{2}$$" ], "result": "=uu^{2}-u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u^{2}$$", "result": "=u^{2}\\left(u-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=-\\left(u-1\\right)+u^{2}\\left(u-1\\right)" }, { "type": "step", "primary": "Factor out common term $$u-1$$", "result": "=\\left(u-1\\right)\\left(u^{2}-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "interim", "title": "Factor $$u^{2}-1:{\\quad}\\left(u+1\\right)\\left(u-1\\right)$$", "input": "u^{2}-1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=u^{2}-1^{2}" }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$u^{2}-1^{2}=\\left(u+1\\right)\\left(u-1\\right)$$" ], "result": "=\\left(u+1\\right)\\left(u-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\left(u-1\\right)\\left(u+1\\right)\\left(u-1\\right)" }, { "type": "step", "primary": "Refine", "result": "=\\left(u-1\\right)^{2}\\left(u+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "\\left(u-1\\right)^{2}\\left(u+1\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "u-1=0\\lor\\:u+1=0" }, { "type": "interim", "title": "Solve $$u-1=0:{\\quad}u=1$$", "input": "u-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "u-1=0", "result": "u=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "u-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "u=1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$u+1=0:{\\quad}u=-1$$", "input": "u+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "u+1=0", "result": "u=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "u+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "u=-1" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "u=1,\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(2x\\right)$$", "result": "\\tan\\left(2x\\right)=1,\\:\\tan\\left(2x\\right)=-1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\tan\\left(2x\\right)=1{\\quad:\\quad}x=\\frac{π}{8}+\\frac{πn}{2}$$", "input": "\\tan\\left(2x\\right)=1", "steps": [ { "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": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\tan\\left(2x\\right)=-1{\\quad:\\quad}x=\\frac{3π}{8}+\\frac{πn}{2}$$", "input": "\\tan\\left(2x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(2x\\right)=-1$$", "result": "2x=\\frac{3π}{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{3π}{4}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$2x=\\frac{3π}{4}+πn:{\\quad}x=\\frac{3π}{8}+\\frac{πn}{2}$$", "input": "2x=\\frac{3π}{4}+πn", "steps": [ { "type": "interim", "title": "Divide both sides by $$2$$", "input": "2x=\\frac{3π}{4}+πn", "result": "x=\\frac{3π}{8}+\\frac{πn}{2}", "steps": [ { "type": "step", "primary": "Divide both sides by $$2$$", "result": "\\frac{2x}{2}=\\frac{\\frac{3π}{4}}{2}+\\frac{πn}{2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{2x}{2}=\\frac{\\frac{3π}{4}}{2}+\\frac{πn}{2}", "result": "x=\\frac{3π}{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 $$\\frac{\\frac{3π}{4}}{2}+\\frac{πn}{2}:{\\quad}\\frac{3π}{8}+\\frac{πn}{2}$$", "input": "\\frac{\\frac{3π}{4}}{2}+\\frac{πn}{2}", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{3π}{4}}{2}=\\frac{3π}{8}$$", "input": "\\frac{\\frac{3π}{4}}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{\\frac{b}{c}}{a}=\\frac{b}{c\\:\\cdot\\:a}$$", "result": "=\\frac{3π}{4\\cdot\\:2}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:2=8$$", "result": "=\\frac{3π}{8}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajb/b0aK6mxzkrRQ3Y2KdbDjdd47a0hQ8flDbGsI5To1drfMRPynLSfP3VYe1OQldRK0Yo9KYWhFtMkDP9QQgDmMWtOub2dAxgaWacwKjDMkB6iXzgR8rXBT7/gCovYjN/VysbmH6V1a/M1Gp5krfeRUkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=\\frac{3π}{8}+\\frac{πn}{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajTSexn9aeRxnCqKa8zEDxn2MEOV06NabtIQ9KVgA6uqjzRqDxPUzBN6vjj5oJL9kUP6VElgM3HMMs7DP5SaP9shcacZaSnYho+jSItR5D3mEFRT+Wj6fEoaZz5e+WjdB4pjcBIL5pmo83UMFZRSzJsUt8Vz4ri5I/oyZLRAY9v9SQKn6Q4qESPWLSgGE8bEuS/gic6CeEliVmNlf8MSKF6e/Mg94S0N9we//Py6WzxN6" } }, { "type": "step", "result": "x=\\frac{3π}{8}+\\frac{πn}{2}" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7IPxU0Sz5OsfkKDEf2pw7GysrCMgqAu4vLSYhf00Xv1WTdaV09PMxEKZ9FieghTFwYDSeuxDg/JbzQA94n1cRKOgoRPK/2Tf/pGvcKW+heVYpv24EdgpEixUChcD9oAcIXGnGWkp2IaPo0iLUeQ95hHsI7YL9j/yxi5GNEDafsz7PbvGmKJwIuV6P6vcqqmDqs6cMsPSOyx1UIMZhE+2hVBt2OQeQFJaFtymbvC2WXqw23P//xmD2+Up6/J3QUrQLSuEuvolv4v3JoYWEkFTH1yWcYSkJ17IY8FE++yIuawu3tnrGmRXRcdoZ+5k2AgF14IMIfYNwM2JqUy6/LsGfCeT5Rj0kETm/9hJnqz4ZzPOdiqcAK81Ko4/KpyFPiCZ8dptnCLknK03/W5+ZCJKKaQYyBJL2j075R6FHVrdPUhNadgwzuY6Jb0cogSIeKVquUcbinK14Gon5f3QBcn/JKpwBSyQ/f/TDfHNs9qNDLfIpcZPZm+pU/lbBV9einQehfhvb9mimrAoYFeuVVvHE1g==" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=\\frac{3π}{8}+\\frac{πn}{2}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{π}{8}+\\frac{πn}{2},\\:x=\\frac{3π}{8}+\\frac{πn}{2}" } ], "meta": { "solvingClass": "Trig Equations", "practiceLink": "/practice/trigonometry-practice#area=main&subtopic=Trig%20Equations", "practiceTopic": "Trig Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "\\tan^{3}(2x)-\\tan^{2}(2x)-\\tan(2x)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }