sin^3(x)=-1

{ "query": { "display": "$$\\sin^{3}\\left(x\\right)=-1$$", "symbolab_question": "EQUATION#\\sin^{3}(x)=-1" }, "solution": { "level": "PERFORMED", "subject": "Trigonometry", "topic": "Trig Equations", "subTopic": "Trig Equations", "default": "x=\\frac{3π}{2}+2πn", "degrees": "x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{3}\\left(x\\right)=-1{\\quad:\\quad}x=\\frac{3π}{2}+2πn$$", "input": "\\sin^{3}\\left(x\\right)=-1", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{3}\\left(x\\right)=-1", "result": "\\sin\\left(x\\right)=-1,\\:\\sin\\left(x\\right)=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2},\\:\\sin\\left(x\\right)=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{3}=-1" }, { "type": "interim", "title": "$$u^{3}=-1{\\quad:\\quad}u=-1,\\:u=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2},\\:u=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}$$", "input": "u^{3}=-1", "steps": [ { "type": "step", "primary": "For $$x^{3}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt[3]{f\\left(a\\right)},\\:\\sqrt[3]{f\\left(a\\right)}\\frac{-1-\\sqrt{3}i}{2},\\:\\sqrt[3]{f\\left(a\\right)}\\frac{-1+\\sqrt{3}i}{2}$$" }, { "type": "step", "result": "u=\\sqrt[3]{-1},\\:u=\\sqrt[3]{-1}\\frac{-1+\\sqrt{3}i}{2},\\:u=\\sqrt[3]{-1}\\frac{-1-\\sqrt{3}i}{2}" }, { "type": "interim", "title": "$$\\sqrt[3]{-1}=-1$$", "input": "\\sqrt[3]{-1}", "steps": [ { "type": "step", "primary": "$$\\sqrt[n]{-1}=-1,\\:$$if $$n$$ is odd", "secondary": [ "$$\\sqrt[3]{-1}=-1$$" ], "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76H1xoeptwwOonZRztLDv7i061ljBSPJeENOw2efoSWuixCfF7o9L2hRx9Shm0jmvo3oe/oyhMy2+1TQhDBd2f8qdr6Xp5CVvDj5nIWgDseQkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "interim", "title": "Simplify $$\\sqrt[3]{-1}\\frac{-1+\\sqrt{3}i}{2}:{\\quad}\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt[3]{-1}\\frac{-1+\\sqrt{3}i}{2}", "steps": [ { "type": "interim", "title": "$$\\sqrt[3]{-1}=-1$$", "input": "\\sqrt[3]{-1}", "steps": [ { "type": "step", "primary": "$$\\sqrt[n]{-1}=-1,\\:$$if $$n$$ is odd", "secondary": [ "$$\\sqrt[3]{-1}=-1$$" ], "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76H1xoeptwwOonZRztLDv7i061ljBSPJeENOw2efoSWuixCfF7o9L2hRx9Shm0jmvo3oe/oyhMy2+1TQhDBd2f8qdr6Xp5CVvDj5nIWgDseQkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-1\\cdot\\:\\frac{-1+\\sqrt{3}i}{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{-1+\\sqrt{3}i}{2}=\\frac{-1+\\sqrt{3}i}{2}$$", "result": "=-\\frac{-1+\\sqrt{3}i}{2}" }, { "type": "interim", "title": "Rewrite $$-\\frac{-1+\\sqrt{3}i}{2}$$ in standard complex form: $$\\frac{1}{2}-\\frac{\\sqrt{3}}{2}i$$", "input": "-\\frac{-1+\\sqrt{3}i}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{-1+\\sqrt{3}i}{2}=-\\left(-\\frac{1}{2}\\right)-\\left(\\frac{\\sqrt{3}i}{2}\\right)$$" ], "result": "=-\\left(-\\frac{1}{2}\\right)-\\left(\\frac{\\sqrt{3}i}{2}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a,\\:-\\left(-a\\right)=a$$", "result": "=\\frac{1}{2}-\\frac{\\sqrt{3}i}{2}" } ], "meta": { "interimType": "Rewrite In Complex Form Title 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74TqdRGDyv3oqElK9EevGS9z4F6b9EBEhpTHCu4QIJPQnVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZwPTM+xnhoGkKkgIQ0VhzmNz4F6b9EBEhpTHCu4QIJPRMYix7/cTjHR2UPxtgKfavQO9djmXGGOQuWkFSRSz08+WvT9lR6dzrjAd/DNDcBv0=" } }, { "type": "step", "result": "=\\frac{1}{2}-\\frac{\\sqrt{3}}{2}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "interim", "title": "Simplify $$\\sqrt[3]{-1}\\frac{-1-\\sqrt{3}i}{2}:{\\quad}\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}$$", "input": "\\sqrt[3]{-1}\\frac{-1-\\sqrt{3}i}{2}", "steps": [ { "type": "interim", "title": "$$\\sqrt[3]{-1}=-1$$", "input": "\\sqrt[3]{-1}", "steps": [ { "type": "step", "primary": "$$\\sqrt[n]{-1}=-1,\\:$$if $$n$$ is odd", "secondary": [ "$$\\sqrt[3]{-1}=-1$$" ], "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76H1xoeptwwOonZRztLDv7i061ljBSPJeENOw2efoSWuixCfF7o9L2hRx9Shm0jmvo3oe/oyhMy2+1TQhDBd2f8qdr6Xp5CVvDj5nIWgDseQkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=-1\\cdot\\:\\frac{-1-\\sqrt{3}i}{2}" }, { "type": "step", "primary": "Multiply: $$1\\cdot\\:\\frac{-1-\\sqrt{3}i}{2}=\\frac{-1-\\sqrt{3}i}{2}$$", "result": "=-\\frac{-1-\\sqrt{3}i}{2}" }, { "type": "interim", "title": "Rewrite $$-\\frac{-1-\\sqrt{3}i}{2}$$ in standard complex form: $$\\frac{1}{2}+\\frac{\\sqrt{3}}{2}i$$", "input": "-\\frac{-1-\\sqrt{3}i}{2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{-1-\\sqrt{3}i}{2}=-\\left(-\\frac{1}{2}\\right)-\\left(-\\frac{\\sqrt{3}i}{2}\\right)$$" ], "result": "=-\\left(-\\frac{1}{2}\\right)-\\left(-\\frac{\\sqrt{3}i}{2}\\right)" }, { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\frac{1}{2}+\\frac{\\sqrt{3}i}{2}" } ], "meta": { "interimType": "Rewrite In Complex Form Title 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74TqdRGDyv3oqElK9EevGS1BGfus/0qXQoc6ObyzUa30nVQZ9jTQhWaoOGaFYelMKfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZwPTM+xnhoGkKkgIQ0VhzmFBGfus/0qXQoc6ObyzUa31MYix7/cTjHR2UPxtgKfavQO9djmXGGOQuWkFSRSz08+WvT9lR6dzrjAd/DNDcBv0=" } }, { "type": "step", "result": "=\\frac{1}{2}+\\frac{\\sqrt{3}}{2}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "u=-1,\\:u=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2},\\:u=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}" } ], "meta": { "solvingClass": "Equations", "interimType": "Equations" } }, { "type": "step", "primary": "Substitute back $$u=\\sin\\left(x\\right)$$", "result": "\\sin\\left(x\\right)=-1,\\:\\sin\\left(x\\right)=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2},\\:\\sin\\left(x\\right)=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}" } ], "meta": { "interimType": "Substitution Method 0Eq" } }, { "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": "$$\\sin\\left(x\\right)=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=\\frac{1}{2}-i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\sin\\left(x\\right)=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}{\\quad:\\quad}$$No Solution", "input": "\\sin\\left(x\\right)=\\frac{1}{2}+i\\frac{\\sqrt{3}}{2}", "steps": [ { "type": "step", "result": "\\mathrm{No\\:Solution}" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Combine all the solutions", "result": "x=\\frac{3π}{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)+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }