tan^3(x)-2tan(x)=tan(x)

{ "query": { "display": "$$\\tan^{3}\\left(x\\right)-2\\tan\\left(x\\right)=\\tan\\left(x\\right)$$", "symbolab_question": "EQUATION#\\tan^{3}(x)-2\\tan(x)=\\tan(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=πn,x=\\frac{2π}{3}+πn,x=\\frac{π}{3}+πn", "degrees": "x=0^{\\circ }+180^{\\circ }n,x=120^{\\circ }+180^{\\circ }n,x=60^{\\circ }+180^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\tan^{3}\\left(x\\right)-2\\tan\\left(x\\right)=\\tan\\left(x\\right){\\quad:\\quad}x=πn,\\:x=\\frac{2π}{3}+πn,\\:x=\\frac{π}{3}+πn$$", "input": "\\tan^{3}\\left(x\\right)-2\\tan\\left(x\\right)=\\tan\\left(x\\right)", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\tan^{3}\\left(x\\right)-2\\tan\\left(x\\right)=\\tan\\left(x\\right)", "result": "\\tan\\left(x\\right)=0,\\:\\tan\\left(x\\right)=-\\sqrt{3},\\:\\tan\\left(x\\right)=\\sqrt{3}", "steps": [ { "type": "step", "primary": "Let: $$\\tan\\left(x\\right)=u$$", "result": "u^{3}-2u=u" }, { "type": "interim", "title": "$$u^{3}-2u=u{\\quad:\\quad}u=0,\\:u=-\\sqrt{3},\\:u=\\sqrt{3}$$", "input": "u^{3}-2u=u", "steps": [ { "type": "interim", "title": "Move $$u\\:$$to the left side", "input": "u^{3}-2u=u", "result": "u^{3}-3u=0", "steps": [ { "type": "step", "primary": "Subtract $$u$$ from both sides", "result": "u^{3}-2u-u=u-u" }, { "type": "step", "primary": "Simplify", "result": "u^{3}-3u=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Factor $$u^{3}-3u:{\\quad}u\\left(u+\\sqrt{3}\\right)\\left(u-\\sqrt{3}\\right)$$", "input": "u^{3}-3u", "steps": [ { "type": "interim", "title": "Factor out common term $$u:{\\quad}u\\left(u^{2}-3\\right)$$", "input": "u^{3}-3u", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{3}=u^{2}u$$" ], "result": "=u^{2}u-3u", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u$$", "result": "=u\\left(u^{2}-3\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=u\\left(u^{2}-3\\right)" }, { "type": "interim", "title": "Factor $$u^{2}-3:{\\quad}\\left(u+\\sqrt{3}\\right)\\left(u-\\sqrt{3}\\right)$$", "input": "u^{2}-3", "steps": [ { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$3=\\left(\\sqrt{3}\\right)^{2}$$" ], "result": "=u^{2}-\\left(\\sqrt{3}\\right)^{2}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$u^{2}-\\left(\\sqrt{3}\\right)^{2}=\\left(u+\\sqrt{3}\\right)\\left(u-\\sqrt{3}\\right)$$" ], "result": "=\\left(u+\\sqrt{3}\\right)\\left(u-\\sqrt{3}\\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": "=u\\left(u+\\sqrt{3}\\right)\\left(u-\\sqrt{3}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "u\\left(u+\\sqrt{3}\\right)\\left(u-\\sqrt{3}\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "u=0\\lor\\:u+\\sqrt{3}=0\\lor\\:u-\\sqrt{3}=0" }, { "type": "interim", "title": "Solve $$u+\\sqrt{3}=0:{\\quad}u=-\\sqrt{3}$$", "input": "u+\\sqrt{3}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{3}\\:$$to the right side", "input": "u+\\sqrt{3}=0", "result": "u=-\\sqrt{3}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{3}$$ from both sides", "result": "u+\\sqrt{3}-\\sqrt{3}=0-\\sqrt{3}" }, { "type": "step", "primary": "Simplify", "result": "u=-\\sqrt{3}" } ], "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 $$u-\\sqrt{3}=0:{\\quad}u=\\sqrt{3}$$", "input": "u-\\sqrt{3}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{3}\\:$$to the right side", "input": "u-\\sqrt{3}=0", "result": "u=\\sqrt{3}", "steps": [ { "type": "step", "primary": "Add $$\\sqrt{3}$$ to both sides", "result": "u-\\sqrt{3}+\\sqrt{3}=0+\\sqrt{3}" }, { "type": "step", "primary": "Simplify", "result": "u=\\sqrt{3}" } ], "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=0,\\:u=-\\sqrt{3},\\:u=\\sqrt{3}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\tan\\left(x\\right)$$", "result": "\\tan\\left(x\\right)=0,\\:\\tan\\left(x\\right)=-\\sqrt{3},\\:\\tan\\left(x\\right)=\\sqrt{3}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\tan\\left(x\\right)=0{\\quad:\\quad}x=πn$$", "input": "\\tan\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(x\\right)=0$$", "result": "x=0+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": "x=0+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+πn:{\\quad}x=πn$$", "input": "x=0+πn", "steps": [ { "type": "step", "primary": "$$0+πn=πn$$", "result": "x=πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\tan\\left(x\\right)=-\\sqrt{3}{\\quad:\\quad}x=\\frac{2π}{3}+πn$$", "input": "\\tan\\left(x\\right)=-\\sqrt{3}", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(x\\right)=-\\sqrt{3}$$", "result": "x=\\frac{2π}{3}+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": "x=\\frac{2π}{3}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\tan\\left(x\\right)=\\sqrt{3}{\\quad:\\quad}x=\\frac{π}{3}+πn$$", "input": "\\tan\\left(x\\right)=\\sqrt{3}", "steps": [ { "type": "interim", "title": "General solutions for $$\\tan\\left(x\\right)=\\sqrt{3}$$", "result": "x=\\frac{π}{3}+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": "x=\\frac{π}{3}+πn" } ], "meta": { "interimType": "Trig General Solutions tan 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=πn,\\:x=\\frac{2π}{3}+πn,\\:x=\\frac{π}{3}+πn" } ], "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}(x)-2\\tan(x)-\\tan(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }