sin^3(x)-2sin(x)=0

{ "query": { "display": "$$\\sin^{3}\\left(x\\right)-2\\sin\\left(x\\right)=0$$", "symbolab_question": "EQUATION#\\sin^{3}(x)-2\\sin(x)=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{3}\\left(x\\right)-2\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin^{3}\\left(x\\right)-2\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{3}\\left(x\\right)-2\\sin\\left(x\\right)=0", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=-\\sqrt{2},\\:\\sin\\left(x\\right)=\\sqrt{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{3}-2u=0" }, { "type": "interim", "title": "$$u^{3}-2u=0{\\quad:\\quad}u=0,\\:u=-\\sqrt{2},\\:u=\\sqrt{2}$$", "input": "u^{3}-2u=0", "steps": [ { "type": "interim", "title": "Factor $$u^{3}-2u:{\\quad}u\\left(u+\\sqrt{2}\\right)\\left(u-\\sqrt{2}\\right)$$", "input": "u^{3}-2u", "steps": [ { "type": "interim", "title": "Factor out common term $$u:{\\quad}u\\left(u^{2}-2\\right)$$", "input": "u^{3}-2u", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{3}=u^{2}u$$" ], "result": "=u^{2}u-2u", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u$$", "result": "=u\\left(u^{2}-2\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=u\\left(u^{2}-2\\right)" }, { "type": "interim", "title": "Factor $$u^{2}-2:{\\quad}\\left(u+\\sqrt{2}\\right)\\left(u-\\sqrt{2}\\right)$$", "input": "u^{2}-2", "steps": [ { "type": "step", "primary": "Apply radical rule: $$a=\\left(\\sqrt{a}\\right)^{2}$$", "secondary": [ "$$2=\\left(\\sqrt{2}\\right)^{2}$$" ], "result": "=u^{2}-\\left(\\sqrt{2}\\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": [ "$$u^{2}-\\left(\\sqrt{2}\\right)^{2}=\\left(u+\\sqrt{2}\\right)\\left(u-\\sqrt{2}\\right)$$" ], "result": "=\\left(u+\\sqrt{2}\\right)\\left(u-\\sqrt{2}\\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{2}\\right)\\left(u-\\sqrt{2}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "u\\left(u+\\sqrt{2}\\right)\\left(u-\\sqrt{2}\\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{2}=0\\lor\\:u-\\sqrt{2}=0" }, { "type": "interim", "title": "Solve $$u+\\sqrt{2}=0:{\\quad}u=-\\sqrt{2}$$", "input": "u+\\sqrt{2}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{2}\\:$$to the right side", "input": "u+\\sqrt{2}=0", "result": "u=-\\sqrt{2}", "steps": [ { "type": "step", "primary": "Subtract $$\\sqrt{2}$$ from both sides", "result": "u+\\sqrt{2}-\\sqrt{2}=0-\\sqrt{2}" }, { "type": "step", "primary": "Simplify", "result": "u=-\\sqrt{2}" } ], "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 $$u-\\sqrt{2}=0:{\\quad}u=\\sqrt{2}$$", "input": "u-\\sqrt{2}=0", "steps": [ { "type": "interim", "title": "Move $$\\sqrt{2}\\:$$to the right side", "input": "u-\\sqrt{2}=0", "result": "u=\\sqrt{2}", "steps": [ { "type": "step", "primary": "Add $$\\sqrt{2}$$ to both sides", "result": "u-\\sqrt{2}+\\sqrt{2}=0+\\sqrt{2}" }, { "type": "step", "primary": "Simplify", "result": "u=\\sqrt{2}" } ], "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{2},\\:u=\\sqrt{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=-\\sqrt{2},\\:\\sin\\left(x\\right)=\\sqrt{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=0{\\quad:\\quad}x=2πn,\\:x=π+2πn$$", "input": "\\sin\\left(x\\right)=0", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=0$$", "result": "x=0+2πn,\\:x=π+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=0+2πn,\\:x=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$x=0+2πn:{\\quad}x=2πn$$", "input": "x=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "x=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "x=2πn,\\:x=π+2πn" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-\\sqrt{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=-\\sqrt{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\sqrt{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=\\sqrt{2}", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+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^{3}(x)-2\\sin(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }