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

{ "query": { "display": "$$\\sin^{3}\\left(x\\right)=\\sin^{2}\\left(x\\right)$$", "symbolab_question": "EQUATION#\\sin^{3}(x)=\\sin^{2}(x)" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=2πn,x=π+2πn,x=\\frac{π}{2}+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n,x=90^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{3}\\left(x\\right)=\\sin^{2}\\left(x\\right){\\quad:\\quad}x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{2}+2πn$$", "input": "\\sin^{3}\\left(x\\right)=\\sin^{2}\\left(x\\right)", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{3}\\left(x\\right)=\\sin^{2}\\left(x\\right)", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{3}=u^{2}" }, { "type": "interim", "title": "$$u^{3}=u^{2}{\\quad:\\quad}u=0,\\:u=1$$", "input": "u^{3}=u^{2}", "steps": [ { "type": "interim", "title": "Move $$u^{2}\\:$$to the left side", "input": "u^{3}=u^{2}", "result": "u^{3}-u^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$u^{2}$$ from both sides", "result": "u^{3}-u^{2}=u^{2}-u^{2}" }, { "type": "step", "primary": "Simplify", "result": "u^{3}-u^{2}=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Factor $$u^{3}-u^{2}:{\\quad}u^{2}\\left(u-1\\right)$$", "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": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "u^{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=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": "step", "primary": "The solutions are", "result": "u=0,\\:u=1" } ], "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)=1" } ], "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": "$$\\sin\\left(x\\right)=1{\\quad:\\quad}x=\\frac{π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=1", "steps": [ { "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": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\: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": "\\sin^{3}(x)-\\sin^{2}(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }