sin^3(x)-6sin^2(x)-sin(x)+6=0

{ "query": { "display": "$$\\sin^{3}\\left(x\\right)-6\\sin^{2}\\left(x\\right)-\\sin\\left(x\\right)+6=0$$", "symbolab_question": "EQUATION#\\sin^{3}(x)-6\\sin^{2}(x)-\\sin(x)+6=0" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{3π}{2}+2πn,x=\\frac{π}{2}+2πn", "degrees": "x=270^{\\circ }+360^{\\circ }n,x=90^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{3}\\left(x\\right)-6\\sin^{2}\\left(x\\right)-\\sin\\left(x\\right)+6=0{\\quad:\\quad}x=\\frac{3π}{2}+2πn,\\:x=\\frac{π}{2}+2πn$$", "input": "\\sin^{3}\\left(x\\right)-6\\sin^{2}\\left(x\\right)-\\sin\\left(x\\right)+6=0", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{3}\\left(x\\right)-6\\sin^{2}\\left(x\\right)-\\sin\\left(x\\right)+6=0", "result": "\\sin\\left(x\\right)=6,\\:\\sin\\left(x\\right)=-1,\\:\\sin\\left(x\\right)=1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{3}-6u^{2}-u+6=0" }, { "type": "interim", "title": "$$u^{3}-6u^{2}-u+6=0{\\quad:\\quad}u=6,\\:u=-1,\\:u=1$$", "input": "u^{3}-6u^{2}-u+6=0", "steps": [ { "type": "interim", "title": "Factor $$u^{3}-6u^{2}-u+6:{\\quad}\\left(u-6\\right)\\left(u+1\\right)\\left(u-1\\right)$$", "input": "u^{3}-6u^{2}-u+6", "steps": [ { "type": "step", "result": "=\\left(u^{3}-6u^{2}\\right)+\\left(-u+6\\right)" }, { "type": "interim", "title": "Factor out $$-1\\:$$from $$-u+6:\\quad\\:-\\left(u-6\\right)$$", "input": "-u+6", "steps": [ { "type": "step", "primary": "Factor out common term $$-1$$", "result": "=-\\left(u-6\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7D+0NG8fQ3RgTin9KrRGaMsevWYjE5TzvOKvzjFpIz0IByGqvpVrcovBtOEfrP9G1KGl0c22SM9+MTKrfhkk8IKN6Hv6MoTMtvtU0IQwXdn/LaG2/wGnl6dNw3H9m17XsThnUlbyBEfXJCWlCtkCLAVH0gXKcs/9zaK6nm+b1wvY=" } }, { "type": "interim", "title": "Factor out $$u^{2}\\:$$from $$u^{3}-6u^{2}:\\quad\\:u^{2}\\left(u-6\\right)$$", "input": "u^{3}-6u^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$u^{3}=uu^{2}$$" ], "result": "=uu^{2}-6u^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$u^{2}$$", "result": "=u^{2}\\left(u-6\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out 3Eq" } }, { "type": "step", "result": "=-\\left(u-6\\right)+u^{2}\\left(u-6\\right)" }, { "type": "step", "primary": "Factor out common term $$u-6$$", "result": "=\\left(u-6\\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-6\\right)\\left(u+1\\right)\\left(u-1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "\\left(u-6\\right)\\left(u+1\\right)\\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-6=0\\lor\\:u+1=0\\lor\\:u-1=0" }, { "type": "interim", "title": "Solve $$u-6=0:{\\quad}u=6$$", "input": "u-6=0", "steps": [ { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "u-6=0", "result": "u=6", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "u-6+6=0+6" }, { "type": "step", "primary": "Simplify", "result": "u=6" } ], "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": "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=6,\\:u=-1,\\:u=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=6,\\:\\sin\\left(x\\right)=-1,\\:\\sin\\left(x\\right)=1" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=6{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=6", "steps": [ { "type": "step", "primary": "$$-1\\le\\sin\\left(x\\right)\\le1$$", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=-1{\\quad:\\quad}x=\\frac{3π}{2}+2πn$$", "input": "\\sin\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "General solutions for $$\\sin\\left(x\\right)=-1$$", "result": "x=\\frac{3π}{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{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } } ], "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=\\frac{3π}{2}+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)-6\\sin^{2}(x)-\\sin(x)+6" }, "showViewLarger": true } }, "meta": { "showVerify": true } }