sin^{22}(x)=sin^2(x)

{ "query": { "display": "$$\\sin^{22}\\left(x\\right)=\\sin^{2}\\left(x\\right)$$", "symbolab_question": "EQUATION#\\sin^{22}(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,x=\\frac{3π}{2}+2πn", "degrees": "x=0^{\\circ }+360^{\\circ }n,x=180^{\\circ }+360^{\\circ }n,x=90^{\\circ }+360^{\\circ }n,x=270^{\\circ }+360^{\\circ }n", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\sin^{22}\\left(x\\right)=\\sin^{2}\\left(x\\right){\\quad:\\quad}x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{2}+2πn,\\:x=\\frac{3π}{2}+2πn$$", "input": "\\sin^{22}\\left(x\\right)=\\sin^{2}\\left(x\\right)", "steps": [ { "type": "interim", "title": "Solve by substitution", "input": "\\sin^{22}\\left(x\\right)=\\sin^{2}\\left(x\\right)", "result": "\\sin\\left(x\\right)=0,\\:\\sin\\left(x\\right)=1,\\:\\sin\\left(x\\right)=-1", "steps": [ { "type": "step", "primary": "Let: $$\\sin\\left(x\\right)=u$$", "result": "u^{22}=u^{2}" }, { "type": "interim", "title": "$$u^{22}=u^{2}{\\quad:\\quad}u=0,\\:u=1,\\:u=-1$$", "input": "u^{22}=u^{2}", "steps": [ { "type": "interim", "title": "Move $$u^{2}\\:$$to the left side", "input": "u^{22}=u^{2}", "result": "u^{22}-u^{2}=0", "steps": [ { "type": "step", "primary": "Subtract $$u^{2}$$ from both sides", "result": "u^{22}-u^{2}=u^{2}-u^{2}" }, { "type": "step", "primary": "Simplify", "result": "u^{22}-u^{2}=0" } ], "meta": { "interimType": "Move to the Left Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7L8oAVlMowtBqqiGCzNdZFFZUrxf4UrSN3py9F20U7Y8Vur0Ux2Di0sW3ZKzSHXsVDkBoZXagRHYaFrkhNABIB4CqiacXnRmza3yMD5APjuU1NX8vYY7QuKAPxlI3NDQqW8Fc+U+/w8rp8/Bvh9ntJFLA/qSJDgiGciFOKvGrX3z6XbvNCPP9DNlCKZnYivn8FM/1+nijDQWWEQXaPfEjgE5VwgE8nYO4wvyWdgInQCWvNukq47XvfOg/T1ZFEy+fR67kL76alZKtYS5UvyFL71wFK0XkITnE2AhWY8mg9XOCxGziJ8rVR1NDgfak5wk1TxiuZJF6Oi5XpwWE3BNQwNTcoBjiQMeKUzqvjloxiaFFPoW8imL5Ihzz6jeBHy3Vi126KflIGXPj0T66SV3pmAInWVDUUDBWNHqYNa25MfkOAOWTWL/jIC0Sf89wVXjAo3oe/oyhMy2+1TQhDBd2fxY8SVeHJk21C+Ewc77ADWQgmd/M5vpRoWTWO+gR0MiZ5EEd95TCm58cCS2zmqfAHw==" } }, { "type": "step", "primary": "Rewrite the equation with $$v=u^{2}$$ and $$v^{11}=u^{22}$$", "result": "v^{11}-v=0" }, { "type": "interim", "title": "Solve $$v^{11}-v=0:{\\quad}v=0,\\:v=-1,\\:v=1$$", "input": "v^{11}-v=0", "steps": [ { "type": "interim", "title": "Factor $$v^{11}-v:{\\quad}v\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)$$", "input": "v^{11}-v", "steps": [ { "type": "interim", "title": "Factor out common term $$v:{\\quad}v\\left(v^{10}-1\\right)$$", "input": "v^{11}-v", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$v^{11}=v^{10}v$$" ], "result": "=v^{10}v-v", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$v$$", "result": "=v\\left(v^{10}-1\\right)" } ], "meta": { "interimType": "Factor Take Out Common Term 1Eq", "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } }, { "type": "step", "result": "=v\\left(v^{10}-1\\right)" }, { "type": "interim", "title": "Factor $$v^{10}-1:{\\quad}\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)$$", "input": "v^{10}-1", "steps": [ { "type": "interim", "title": "Rewrite $$v^{10}-1$$ as $$\\left(v^{5}\\right)^{2}-1^{2}$$", "input": "v^{10}-1", "result": "=\\left(v^{5}\\right)^{2}-1^{2}", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{2}$$", "result": "=v^{10}-1^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$a^{bc}=\\left(a^{b}\\right)^{c}$$", "secondary": [ "$$v^{10}=\\left(v^{5}\\right)^{2}$$" ], "result": "=\\left(v^{5}\\right)^{2}-1^{2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "interimType": "Generic Rewrite As Specific 2Eq" } }, { "type": "step", "primary": "Apply Difference of Two Squares Formula: $$x^{2}-y^{2}=\\left(x+y\\right)\\left(x-y\\right)$$", "secondary": [ "$$\\left(v^{5}\\right)^{2}-1^{2}=\\left(v^{5}+1\\right)\\left(v^{5}-1\\right)$$" ], "result": "=\\left(v^{5}+1\\right)\\left(v^{5}-1\\right)", "meta": { "practiceLink": "/practice/factoring-practice#area=main&subtopic=Difference%20of%20Two%20Squares", "practiceTopic": "Factor Difference of Squares" } }, { "type": "interim", "title": "Factor $$v^{5}+1:{\\quad}\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)$$", "input": "v^{5}+1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{5}$$", "result": "=v^{5}+1^{5}" }, { "type": "step", "primary": "Apply factoring rule: $$x^{n}+y^{n}=\\left(x+y\\right)\\left(x^{n-1}-x^{n-2}y+\\:\\dots\\:-\\:xy^{n-2}\\:+\\:y^{n-1}\\right)\\:\\quad\\:\\quad\\:$$n is odd", "secondary": [ "$$v^{5}+1^{5}=\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)$$" ], "result": "=\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)\\left(v^{5}-1\\right)" }, { "type": "interim", "title": "Factor $$v^{5}-1:{\\quad}\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)$$", "input": "v^{5}-1", "steps": [ { "type": "step", "primary": "Rewrite $$1$$ as $$1^{5}$$", "result": "=v^{5}-1^{5}" }, { "type": "step", "primary": "Apply factoring rule: $$x^{n}-y^{n}=\\left(x-y\\right)\\left(x^{n-1}+x^{n-2}y+\\dots+xy^{n-2}y^{n-1}\\right)$$", "secondary": [ "$$v^{5}-1^{5}=\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)$$" ], "result": "=\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=v\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "v\\left(v+1\\right)\\left(v^{4}-v^{3}+v^{2}-v+1\\right)\\left(v-1\\right)\\left(v^{4}+v^{3}+v^{2}+v+1\\right)=0" }, { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$", "result": "v=0\\lor\\:v+1=0\\lor\\:v^{4}-v^{3}+v^{2}-v+1=0\\lor\\:v-1=0\\lor\\:v^{4}+v^{3}+v^{2}+v+1=0" }, { "type": "interim", "title": "Solve $$v+1=0:{\\quad}v=-1$$", "input": "v+1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "v+1=0", "result": "v=-1", "steps": [ { "type": "step", "primary": "Subtract $$1$$ from both sides", "result": "v+1-1=0-1" }, { "type": "step", "primary": "Simplify", "result": "v=-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 $$v^{4}-v^{3}+v^{2}-v+1=0:{\\quad}$$No Solution for $$v\\in\\mathbb{R}$$", "input": "v^{4}-v^{3}+v^{2}-v+1=0", "steps": [ { "type": "interim", "title": "Find one solution for $$v^{4}-v^{3}+v^{2}-v+1=0$$ using Newton-Raphson:$${\\quad}$$No Solution for $$v\\in\\mathbb{R}$$", "input": "v^{4}-v^{3}+v^{2}-v+1=0", "steps": [ { "type": "definition", "title": "Newton-Raphson Approximation Definition", "text": "The Newton-Raphson method uses an iterative process to approach one root of a function<br/>$$x_{n+1}=x_{n}\\:-\\:\\frac{f\\left(x_{n}\\right)}{f^{\\prime}\\left(x_{n}\\right)}$$" }, { "type": "step", "result": "f\\left(v\\right)=v^{4}-v^{3}+v^{2}-v+1" }, { "type": "interim", "title": "Find $$f^{^{\\prime}}\\left(v\\right):{\\quad}4v^{3}-3v^{2}+2v-1$$", "input": "\\frac{d}{dv}\\left(v^{4}-v^{3}+v^{2}-v+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dv}\\left(v^{4}\\right)-\\frac{d}{dv}\\left(v^{3}\\right)+\\frac{d}{dv}\\left(v^{2}\\right)-\\frac{dv}{dv}+\\frac{d}{dv}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{4}\\right)=4v^{3}$$", "input": "\\frac{d}{dv}\\left(v^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4v^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4v^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYh5Z8KARy5JFAdhy+7OJnqWk3hxk9aCfAWodBRxXgUexNanbR+9UUldIZBa55HxWXv8//6/nV5O4fb8Xgwi7maoRk7nr9IDbDGcsZRPmsBYLH0TsIVfuFzSL2wwDW41bGw==" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{3}\\right)=3v^{2}$$", "input": "\\frac{d}{dv}\\left(v^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3v^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3v^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqpXEbeKr37jRuilFzr8ZE6k3hxk9aCfAWodBRxXgUexYAsXL0SggoaWzn1E3qRqh/8//6/nV5O4fb8Xgwi7maoRk7nr9IDbDGcsZRPmsBYLeGPG1dVOtRmH2mekt/+Ztw==" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{2}\\right)=2v$$", "input": "\\frac{d}{dv}\\left(v^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2v^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiHI6i/lNYJouYKcylLF6B2k3hxk9aCfAWodBRxXgUexsoRboyLdWbDdDvojbJb4SkeCBKuYKgaNJ253gLI69U5feCPJC8Uak4mwlsl/8zOjPWUEL+I3n8Z72JloyPMrWQ==" } }, { "type": "interim", "title": "$$\\frac{dv}{dv}=1$$", "input": "\\frac{dv}{dv}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dv}{dv}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgfyUWYtBrfkB3nVU/L275ZjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIcbQV/ioXryDkiJ7F2u2ivzEWF7EOSaCVV24G9vopS2A" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(1\\right)=0$$", "input": "\\frac{d}{dv}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrYlw0sQkDeBfYmbOrQ+t91J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtHFi1sZ1xQhHh3bCtbuI/B" } }, { "type": "step", "result": "=4v^{3}-3v^{2}+2v-1+0" }, { "type": "step", "primary": "Simplify", "result": "=4v^{3}-3v^{2}+2v-1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Generic Find Title 1Eq" } }, { "type": "step", "primary": "Let $$v_{0}=1$$", "secondary": [ "Compute $$v_{n+1}$$ until $$\\Delta\\:v_{n+1}\\:<\\:0.000001$$" ] }, { "type": "interim", "title": "$$v_{1}=0.5{\\quad:\\quad}Δv_{1}=0.5$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{0}\\right)=1^{4}-1^{3}+1^{2}-1+1=1$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{0}\\right)=4\\cdot\\:1^{3}-3\\cdot\\:1^{2}+2\\cdot\\:1-1=2$$", "$$v_{1}=1-\\frac{1}{2}=0.5$$" ], "result": "v_{1}=0.5" }, { "type": "step", "primary": "$$Δv_{1}=\\left|0.5-1\\right|=0.5$$", "result": "Δv_{1}=0.5" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{2}=3.25{\\quad:\\quad}Δv_{2}=2.75$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{1}\\right)=0.5^{4}-0.5^{3}+0.5^{2}-0.5+1=0.6875$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{1}\\right)=4\\cdot\\:0.5^{3}-3\\cdot\\:0.5^{2}+2\\cdot\\:0.5-1=-0.25$$", "$$v_{2}=0.5-\\frac{0.6875}{-0.25}=3.25$$" ], "result": "v_{2}=3.25" }, { "type": "step", "primary": "$$Δv_{2}=\\left|3.25-0.5\\right|=2.75$$", "result": "Δv_{2}=2.75" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{3}=2.48013…{\\quad:\\quad}Δv_{3}=0.76986…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{2}\\right)=3.25^{4}-3.25^{3}+3.25^{2}-3.25+1=85.55078…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{2}\\right)=4\\cdot\\:3.25^{3}-3\\cdot\\:3.25^{2}+2\\cdot\\:3.25-1=111.125$$", "$$v_{3}=3.25-\\frac{85.55078…}{111.125}=2.48013…$$" ], "result": "v_{3}=2.48013…" }, { "type": "step", "primary": "$$Δv_{3}=\\left|2.48013…-3.25\\right|=0.76986…$$", "result": "Δv_{3}=0.76986…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{4}=1.89445…{\\quad:\\quad}Δv_{4}=0.58568…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{3}\\right)=2.48013…^{4}-2.48013…^{3}+2.48013…^{2}-2.48013…+1=27.25130…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{3}\\right)=4\\cdot\\:2.48013…^{3}-3\\cdot\\:2.48013…^{2}+2\\cdot\\:2.48013…-1=46.52924…$$", "$$v_{4}=2.48013…-\\frac{27.25130…}{46.52924…}=1.89445…$$" ], "result": "v_{4}=1.89445…" }, { "type": "step", "primary": "$$Δv_{4}=\\left|1.89445…-2.48013…\\right|=0.58568…$$", "result": "Δv_{4}=0.58568…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{5}=1.43781…{\\quad:\\quad}Δv_{5}=0.45664…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{4}\\right)=1.89445…^{4}-1.89445…^{3}+1.89445…^{2}-1.89445…+1=8.77607…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{4}\\right)=4\\cdot\\:1.89445…^{3}-3\\cdot\\:1.89445…^{2}+2\\cdot\\:1.89445…-1=19.21862…$$", "$$v_{5}=1.89445…-\\frac{8.77607…}{19.21862…}=1.43781…$$" ], "result": "v_{5}=1.43781…" }, { "type": "step", "primary": "$$Δv_{5}=\\left|1.43781…-1.89445…\\right|=0.45664…$$", "result": "Δv_{5}=0.45664…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{6}=1.05030…{\\quad:\\quad}Δv_{6}=0.38750…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{5}\\right)=1.43781…^{4}-1.43781…^{3}+1.43781…^{2}-1.43781…+1=2.93085…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{5}\\right)=4\\cdot\\:1.43781…^{3}-3\\cdot\\:1.43781…^{2}+2\\cdot\\:1.43781…-1=7.56332…$$", "$$v_{6}=1.43781…-\\frac{2.93085…}{7.56332…}=1.05030…$$" ], "result": "v_{6}=1.05030…" }, { "type": "step", "primary": "$$Δv_{6}=\\left|1.05030…-1.43781…\\right|=0.38750…$$", "result": "Δv_{6}=0.38750…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{7}=0.59224…{\\quad:\\quad}Δv_{7}=0.45805…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{6}\\right)=1.05030…^{4}-1.05030…^{3}+1.05030…^{2}-1.05030…+1=1.11112…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{6}\\right)=4\\cdot\\:1.05030…^{3}-3\\cdot\\:1.05030…^{2}+2\\cdot\\:1.05030…-1=2.42572…$$", "$$v_{7}=1.05030…-\\frac{1.11112…}{2.42572…}=0.59224…$$" ], "result": "v_{7}=0.59224…" }, { "type": "step", "primary": "$$Δv_{7}=\\left|0.59224…-1.05030…\\right|=0.45805…$$", "result": "Δv_{7}=0.45805…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{8}=18.88435…{\\quad:\\quad}Δv_{8}=18.29210…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{7}\\right)=0.59224…^{4}-0.59224…^{3}+0.59224…^{2}-0.59224…+1=0.67380…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{7}\\right)=4\\cdot\\:0.59224…^{3}-3\\cdot\\:0.59224…^{2}+2\\cdot\\:0.59224…-1=-0.03683…$$", "$$v_{8}=0.59224…-\\frac{0.67380…}{-0.03683…}=18.88435…$$" ], "result": "v_{8}=18.88435…" }, { "type": "step", "primary": "$$Δv_{8}=\\left|18.88435…-0.59224…\\right|=18.29210…$$", "result": "Δv_{8}=18.29210…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{9}=14.22188…{\\quad:\\quad}Δv_{9}=4.66247…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{8}\\right)=18.88435…^{4}-18.88435…^{3}+18.88435…^{2}-18.88435…+1=120781.11894…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{8}\\right)=4\\cdot\\:18.88435…^{3}-3\\cdot\\:18.88435…^{2}+2\\cdot\\:18.88435…-1=25904.96293…$$", "$$v_{9}=18.88435…-\\frac{120781.11894…}{25904.96293…}=14.22188…$$" ], "result": "v_{9}=14.22188…" }, { "type": "step", "primary": "$$Δv_{9}=\\left|14.22188…-18.88435…\\right|=4.66247…$$", "result": "Δv_{9}=4.66247…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{10}=10.72385…{\\quad:\\quad}Δv_{10}=3.49802…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{9}\\right)=14.22188…^{4}-14.22188…^{3}+14.22188…^{2}-14.22188…+1=38222.36483…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{9}\\right)=4\\cdot\\:14.22188…^{3}-3\\cdot\\:14.22188…^{2}+2\\cdot\\:14.22188…-1=10926.83534…$$", "$$v_{10}=14.22188…-\\frac{38222.36483…}{10926.83534…}=10.72385…$$" ], "result": "v_{10}=10.72385…" }, { "type": "step", "primary": "$$Δv_{10}=\\left|10.72385…-14.22188…\\right|=3.49802…$$", "result": "Δv_{10}=3.49802…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{11}=8.09884…{\\quad:\\quad}Δv_{11}=2.62501…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{10}\\right)=10.72385…^{4}-10.72385…^{3}+10.72385…^{2}-10.72385…+1=12097.26043…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{10}\\right)=4\\cdot\\:10.72385…^{3}-3\\cdot\\:10.72385…^{2}+2\\cdot\\:10.72385…-1=4608.46147…$$", "$$v_{11}=10.72385…-\\frac{12097.26043…}{4608.46147…}=8.09884…$$" ], "result": "v_{11}=8.09884…" }, { "type": "step", "primary": "$$Δv_{11}=\\left|8.09884…-10.72385…\\right|=2.62501…$$", "result": "Δv_{11}=2.62501…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{12}=6.12820…{\\quad:\\quad}Δv_{12}=1.97063…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{11}\\right)=8.09884…^{4}-8.09884…^{3}+8.09884…^{2}-8.09884…+1=3829.49182…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{11}\\right)=4\\cdot\\:8.09884…^{3}-3\\cdot\\:8.09884…^{2}+2\\cdot\\:8.09884…-1=1943.27695…$$", "$$v_{12}=8.09884…-\\frac{3829.49182…}{1943.27695…}=6.12820…$$" ], "result": "v_{12}=6.12820…" }, { "type": "step", "primary": "$$Δv_{12}=\\left|6.12820…-8.09884…\\right|=1.97063…$$", "result": "Δv_{12}=1.97063…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{13}=4.64785…{\\quad:\\quad}Δv_{13}=1.48034…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{12}\\right)=6.12820…^{4}-6.12820…^{3}+6.12820…^{2}-6.12820…+1=1212.65418…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{12}\\right)=4\\cdot\\:6.12820…^{3}-3\\cdot\\:6.12820…^{2}+2\\cdot\\:6.12820…-1=819.16882…$$", "$$v_{13}=6.12820…-\\frac{1212.65418…}{819.16882…}=4.64785…$$" ], "result": "v_{13}=4.64785…" }, { "type": "step", "primary": "$$Δv_{13}=\\left|4.64785…-6.12820…\\right|=1.48034…$$", "result": "Δv_{13}=1.48034…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{14}=3.53453…{\\quad:\\quad}Δv_{14}=1.11332…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{13}\\right)=4.64785…^{4}-4.64785…^{3}+4.64785…^{2}-4.64785…+1=384.22115…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{13}\\right)=4\\cdot\\:4.64785…^{3}-3\\cdot\\:4.64785…^{2}+2\\cdot\\:4.64785…-1=345.11129…$$", "$$v_{14}=4.64785…-\\frac{384.22115…}{345.11129…}=3.53453…$$" ], "result": "v_{14}=3.53453…" }, { "type": "step", "primary": "$$Δv_{14}=\\left|3.53453…-4.64785…\\right|=1.11332…$$", "result": "Δv_{14}=1.11332…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{15}=2.69527…{\\quad:\\quad}Δv_{15}=0.83926…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{14}\\right)=3.53453…^{4}-3.53453…^{3}+3.53453…^{2}-3.53453…+1=121.87505…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{14}\\right)=4\\cdot\\:3.53453…^{3}-3\\cdot\\:3.53453…^{2}+2\\cdot\\:3.53453…-1=145.21705…$$", "$$v_{15}=3.53453…-\\frac{121.87505…}{145.21705…}=2.69527…$$" ], "result": "v_{15}=2.69527…" }, { "type": "step", "primary": "$$Δv_{15}=\\left|2.69527…-3.53453…\\right|=0.83926…$$", "result": "Δv_{15}=0.83926…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{16}=2.05895…{\\quad:\\quad}Δv_{16}=0.63632…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{15}\\right)=2.69527…^{4}-2.69527…^{3}+2.69527…^{2}-2.69527…+1=38.76232…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{15}\\right)=4\\cdot\\:2.69527…^{3}-3\\cdot\\:2.69527…^{2}+2\\cdot\\:2.69527…-1=60.91625…$$", "$$v_{16}=2.69527…-\\frac{38.76232…}{60.91625…}=2.05895…$$" ], "result": "v_{16}=2.05895…" }, { "type": "step", "primary": "$$Δv_{16}=\\left|2.05895…-2.69527…\\right|=0.63632…$$", "result": "Δv_{16}=0.63632…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{17}=1.56818…{\\quad:\\quad}Δv_{17}=0.49077…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{16}\\right)=2.05895…^{4}-2.05895…^{3}+2.05895…^{2}-2.05895…+1=12.42335…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{16}\\right)=4\\cdot\\:2.05895…^{3}-3\\cdot\\:2.05895…^{2}+2\\cdot\\:2.05895…-1=25.31395…$$", "$$v_{17}=2.05895…-\\frac{12.42335…}{25.31395…}=1.56818…$$" ], "result": "v_{17}=1.56818…" }, { "type": "step", "primary": "$$Δv_{17}=\\left|1.56818…-2.05895…\\right|=0.49077…$$", "result": "Δv_{17}=0.49077…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{18}=1.16736…{\\quad:\\quad}Δv_{18}=0.40081…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{17}\\right)=1.56818…^{4}-1.56818…^{3}+1.56818…^{2}-1.56818…+1=4.08216…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{17}\\right)=4\\cdot\\:1.56818…^{3}-3\\cdot\\:1.56818…^{2}+2\\cdot\\:1.56818…-1=10.18459…$$", "$$v_{18}=1.56818…-\\frac{4.08216…}{10.18459…}=1.16736…$$" ], "result": "v_{18}=1.16736…" }, { "type": "step", "primary": "$$Δv_{18}=\\left|1.16736…-1.56818…\\right|=0.40081…$$", "result": "Δv_{18}=0.40081…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{19}=0.76245…{\\quad:\\quad}Δv_{19}=0.40490…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{18}\\right)=1.16736…^{4}-1.16736…^{3}+1.16736…^{2}-1.16736…+1=1.46161…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{18}\\right)=4\\cdot\\:1.16736…^{3}-3\\cdot\\:1.16736…^{2}+2\\cdot\\:1.16736…-1=3.60974…$$", "$$v_{19}=1.16736…-\\frac{1.46161…}{3.60974…}=0.76245…$$" ], "result": "v_{19}=0.76245…" }, { "type": "step", "primary": "$$Δv_{19}=\\left|0.76245…-1.16736…\\right|=0.40490…$$", "result": "Δv_{19}=0.40490…" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Cannot find solution" } ], "meta": { "interimType": "Newton Raphson Find Real Solution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjALZJIPfVQGIeIQV5JvpOGUA9ycmms1z38QBEuFiVL5aUQWTx3vXy+3rJkf5QRUUlWanAX5CivPcsI6uQ4LY8E+VPFsSrYxiR7WqQy7NlDscd/JUJRroD486Rh7zIYdtR/AV95ouxYbhT9hmx3c2nAq1sD7NfhsPe7eDHrmjY0mE8ROzzRiV/WzHpTRU/mmLvY688IB3iI1IPsojE1WIu/GgF8Wo7UWZyPJl2rEkxDhwEBbPSc7ROvw/06D3ESKqojNHfOwYnV7aMLeoXuGfXFjvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "primary": "The solution is", "result": "\\mathrm{No\\:Solution\\:for}\\:v\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$v-1=0:{\\quad}v=1$$", "input": "v-1=0", "steps": [ { "type": "interim", "title": "Move $$1\\:$$to the right side", "input": "v-1=0", "result": "v=1", "steps": [ { "type": "step", "primary": "Add $$1$$ to both sides", "result": "v-1+1=0+1" }, { "type": "step", "primary": "Simplify", "result": "v=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 $$v^{4}+v^{3}+v^{2}+v+1=0:{\\quad}$$No Solution for $$v\\in\\mathbb{R}$$", "input": "v^{4}+v^{3}+v^{2}+v+1=0", "steps": [ { "type": "interim", "title": "Find one solution for $$v^{4}+v^{3}+v^{2}+v+1=0$$ using Newton-Raphson:$${\\quad}$$No Solution for $$v\\in\\mathbb{R}$$", "input": "v^{4}+v^{3}+v^{2}+v+1=0", "steps": [ { "type": "definition", "title": "Newton-Raphson Approximation Definition", "text": "The Newton-Raphson method uses an iterative process to approach one root of a function<br/>$$x_{n+1}=x_{n}\\:-\\:\\frac{f\\left(x_{n}\\right)}{f^{\\prime}\\left(x_{n}\\right)}$$" }, { "type": "step", "result": "f\\left(v\\right)=v^{4}+v^{3}+v^{2}+v+1" }, { "type": "interim", "title": "Find $$f^{^{\\prime}}\\left(v\\right):{\\quad}4v^{3}+3v^{2}+2v+1$$", "input": "\\frac{d}{dv}\\left(v^{4}+v^{3}+v^{2}+v+1\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dv}\\left(v^{4}\\right)+\\frac{d}{dv}\\left(v^{3}\\right)+\\frac{d}{dv}\\left(v^{2}\\right)+\\frac{dv}{dv}+\\frac{d}{dv}\\left(1\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{4}\\right)=4v^{3}$$", "input": "\\frac{d}{dv}\\left(v^{4}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=4v^{4-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=4v^{3}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYh5Z8KARy5JFAdhy+7OJnqWk3hxk9aCfAWodBRxXgUexNanbR+9UUldIZBa55HxWXv8//6/nV5O4fb8Xgwi7maoRk7nr9IDbDGcsZRPmsBYLH0TsIVfuFzSL2wwDW41bGw==" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{3}\\right)=3v^{2}$$", "input": "\\frac{d}{dv}\\left(v^{3}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=3v^{3-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=3v^{2}", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYqpXEbeKr37jRuilFzr8ZE6k3hxk9aCfAWodBRxXgUexYAsXL0SggoaWzn1E3qRqh/8//6/nV5O4fb8Xgwi7maoRk7nr9IDbDGcsZRPmsBYLeGPG1dVOtRmH2mekt/+Ztw==" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(v^{2}\\right)=2v$$", "input": "\\frac{d}{dv}\\left(v^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2v^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2v", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYiHI6i/lNYJouYKcylLF6B2k3hxk9aCfAWodBRxXgUexsoRboyLdWbDdDvojbJb4SkeCBKuYKgaNJ253gLI69U5feCPJC8Uak4mwlsl/8zOjPWUEL+I3n8Z72JloyPMrWQ==" } }, { "type": "interim", "title": "$$\\frac{dv}{dv}=1$$", "input": "\\frac{dv}{dv}", "steps": [ { "type": "step", "primary": "Apply the common derivative: $$\\frac{dv}{dv}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgfyUWYtBrfkB3nVU/L275ZjqLYrB3CcI0Y7zGHBJCja+8ZDu8iF4MSewt4yms1lIcbQV/ioXryDkiJ7F2u2ivzEWF7EOSaCVV24G9vopS2A" } }, { "type": "interim", "title": "$$\\frac{d}{dv}\\left(1\\right)=0$$", "input": "\\frac{d}{dv}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrYlw0sQkDeBfYmbOrQ+t91J8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTtHFi1sZ1xQhHh3bCtbuI/B" } }, { "type": "step", "result": "=4v^{3}+3v^{2}+2v+1+0" }, { "type": "step", "primary": "Simplify", "result": "=4v^{3}+3v^{2}+2v+1", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Generic Find Title 1Eq" } }, { "type": "step", "primary": "Let $$v_{0}=-1$$", "secondary": [ "Compute $$v_{n+1}$$ until $$\\Delta\\:v_{n+1}\\:<\\:0.000001$$" ] }, { "type": "interim", "title": "$$v_{1}=-0.5{\\quad:\\quad}Δv_{1}=0.5$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{0}\\right)=\\left(-1\\right)^{4}+\\left(-1\\right)^{3}+\\left(-1\\right)^{2}+\\left(-1\\right)+1=1$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{0}\\right)=4\\left(-1\\right)^{3}+3\\left(-1\\right)^{2}+2\\left(-1\\right)+1=-2$$", "$$v_{1}=-1-\\frac{1}{-2}=-0.5$$" ], "result": "v_{1}=-0.5" }, { "type": "step", "primary": "$$Δv_{1}=\\left|-0.5-\\left(-1\\right)\\right|=0.5$$", "result": "Δv_{1}=0.5" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{2}=-3.25{\\quad:\\quad}Δv_{2}=2.75$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{1}\\right)=\\left(-0.5\\right)^{4}+\\left(-0.5\\right)^{3}+\\left(-0.5\\right)^{2}+\\left(-0.5\\right)+1=0.6875$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{1}\\right)=4\\left(-0.5\\right)^{3}+3\\left(-0.5\\right)^{2}+2\\left(-0.5\\right)+1=0.25$$", "$$v_{2}=-0.5-\\frac{0.6875}{0.25}=-3.25$$" ], "result": "v_{2}=-3.25" }, { "type": "step", "primary": "$$Δv_{2}=\\left|-3.25-\\left(-0.5\\right)\\right|=2.75$$", "result": "Δv_{2}=2.75" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{3}=-2.48013…{\\quad:\\quad}Δv_{3}=0.76986…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{2}\\right)=\\left(-3.25\\right)^{4}+\\left(-3.25\\right)^{3}+\\left(-3.25\\right)^{2}+\\left(-3.25\\right)+1=85.55078…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{2}\\right)=4\\left(-3.25\\right)^{3}+3\\left(-3.25\\right)^{2}+2\\left(-3.25\\right)+1=-111.125$$", "$$v_{3}=-3.25-\\frac{85.55078…}{-111.125}=-2.48013…$$" ], "result": "v_{3}=-2.48013…" }, { "type": "step", "primary": "$$Δv_{3}=\\left|-2.48013…-\\left(-3.25\\right)\\right|=0.76986…$$", "result": "Δv_{3}=0.76986…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{4}=-1.89445…{\\quad:\\quad}Δv_{4}=0.58568…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{3}\\right)=\\left(-2.48013…\\right)^{4}+\\left(-2.48013…\\right)^{3}+\\left(-2.48013…\\right)^{2}+\\left(-2.48013…\\right)+1=27.25130…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{3}\\right)=4\\left(-2.48013…\\right)^{3}+3\\left(-2.48013…\\right)^{2}+2\\left(-2.48013…\\right)+1=-46.52924…$$", "$$v_{4}=-2.48013…-\\frac{27.25130…}{-46.52924…}=-1.89445…$$" ], "result": "v_{4}=-1.89445…" }, { "type": "step", "primary": "$$Δv_{4}=\\left|-1.89445…-\\left(-2.48013…\\right)\\right|=0.58568…$$", "result": "Δv_{4}=0.58568…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{5}=-1.43781…{\\quad:\\quad}Δv_{5}=0.45664…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{4}\\right)=\\left(-1.89445…\\right)^{4}+\\left(-1.89445…\\right)^{3}+\\left(-1.89445…\\right)^{2}+\\left(-1.89445…\\right)+1=8.77607…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{4}\\right)=4\\left(-1.89445…\\right)^{3}+3\\left(-1.89445…\\right)^{2}+2\\left(-1.89445…\\right)+1=-19.21862…$$", "$$v_{5}=-1.89445…-\\frac{8.77607…}{-19.21862…}=-1.43781…$$" ], "result": "v_{5}=-1.43781…" }, { "type": "step", "primary": "$$Δv_{5}=\\left|-1.43781…-\\left(-1.89445…\\right)\\right|=0.45664…$$", "result": "Δv_{5}=0.45664…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{6}=-1.05030…{\\quad:\\quad}Δv_{6}=0.38750…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{5}\\right)=\\left(-1.43781…\\right)^{4}+\\left(-1.43781…\\right)^{3}+\\left(-1.43781…\\right)^{2}+\\left(-1.43781…\\right)+1=2.93085…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{5}\\right)=4\\left(-1.43781…\\right)^{3}+3\\left(-1.43781…\\right)^{2}+2\\left(-1.43781…\\right)+1=-7.56332…$$", "$$v_{6}=-1.43781…-\\frac{2.93085…}{-7.56332…}=-1.05030…$$" ], "result": "v_{6}=-1.05030…" }, { "type": "step", "primary": "$$Δv_{6}=\\left|-1.05030…-\\left(-1.43781…\\right)\\right|=0.38750…$$", "result": "Δv_{6}=0.38750…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{7}=-0.59224…{\\quad:\\quad}Δv_{7}=0.45805…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{6}\\right)=\\left(-1.05030…\\right)^{4}+\\left(-1.05030…\\right)^{3}+\\left(-1.05030…\\right)^{2}+\\left(-1.05030…\\right)+1=1.11112…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{6}\\right)=4\\left(-1.05030…\\right)^{3}+3\\left(-1.05030…\\right)^{2}+2\\left(-1.05030…\\right)+1=-2.42572…$$", "$$v_{7}=-1.05030…-\\frac{1.11112…}{-2.42572…}=-0.59224…$$" ], "result": "v_{7}=-0.59224…" }, { "type": "step", "primary": "$$Δv_{7}=\\left|-0.59224…-\\left(-1.05030…\\right)\\right|=0.45805…$$", "result": "Δv_{7}=0.45805…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{8}=-18.88435…{\\quad:\\quad}Δv_{8}=18.29210…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{7}\\right)=\\left(-0.59224…\\right)^{4}+\\left(-0.59224…\\right)^{3}+\\left(-0.59224…\\right)^{2}+\\left(-0.59224…\\right)+1=0.67380…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{7}\\right)=4\\left(-0.59224…\\right)^{3}+3\\left(-0.59224…\\right)^{2}+2\\left(-0.59224…\\right)+1=0.03683…$$", "$$v_{8}=-0.59224…-\\frac{0.67380…}{0.03683…}=-18.88435…$$" ], "result": "v_{8}=-18.88435…" }, { "type": "step", "primary": "$$Δv_{8}=\\left|-18.88435…-\\left(-0.59224…\\right)\\right|=18.29210…$$", "result": "Δv_{8}=18.29210…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{9}=-14.22188…{\\quad:\\quad}Δv_{9}=4.66247…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{8}\\right)=\\left(-18.88435…\\right)^{4}+\\left(-18.88435…\\right)^{3}+\\left(-18.88435…\\right)^{2}+\\left(-18.88435…\\right)+1=120781.11894…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{8}\\right)=4\\left(-18.88435…\\right)^{3}+3\\left(-18.88435…\\right)^{2}+2\\left(-18.88435…\\right)+1=-25904.96293…$$", "$$v_{9}=-18.88435…-\\frac{120781.11894…}{-25904.96293…}=-14.22188…$$" ], "result": "v_{9}=-14.22188…" }, { "type": "step", "primary": "$$Δv_{9}=\\left|-14.22188…-\\left(-18.88435…\\right)\\right|=4.66247…$$", "result": "Δv_{9}=4.66247…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{10}=-10.72385…{\\quad:\\quad}Δv_{10}=3.49802…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{9}\\right)=\\left(-14.22188…\\right)^{4}+\\left(-14.22188…\\right)^{3}+\\left(-14.22188…\\right)^{2}+\\left(-14.22188…\\right)+1=38222.36483…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{9}\\right)=4\\left(-14.22188…\\right)^{3}+3\\left(-14.22188…\\right)^{2}+2\\left(-14.22188…\\right)+1=-10926.83534…$$", "$$v_{10}=-14.22188…-\\frac{38222.36483…}{-10926.83534…}=-10.72385…$$" ], "result": "v_{10}=-10.72385…" }, { "type": "step", "primary": "$$Δv_{10}=\\left|-10.72385…-\\left(-14.22188…\\right)\\right|=3.49802…$$", "result": "Δv_{10}=3.49802…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{11}=-8.09884…{\\quad:\\quad}Δv_{11}=2.62501…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{10}\\right)=\\left(-10.72385…\\right)^{4}+\\left(-10.72385…\\right)^{3}+\\left(-10.72385…\\right)^{2}+\\left(-10.72385…\\right)+1=12097.26043…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{10}\\right)=4\\left(-10.72385…\\right)^{3}+3\\left(-10.72385…\\right)^{2}+2\\left(-10.72385…\\right)+1=-4608.46147…$$", "$$v_{11}=-10.72385…-\\frac{12097.26043…}{-4608.46147…}=-8.09884…$$" ], "result": "v_{11}=-8.09884…" }, { "type": "step", "primary": "$$Δv_{11}=\\left|-8.09884…-\\left(-10.72385…\\right)\\right|=2.62501…$$", "result": "Δv_{11}=2.62501…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{12}=-6.12820…{\\quad:\\quad}Δv_{12}=1.97063…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{11}\\right)=\\left(-8.09884…\\right)^{4}+\\left(-8.09884…\\right)^{3}+\\left(-8.09884…\\right)^{2}+\\left(-8.09884…\\right)+1=3829.49182…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{11}\\right)=4\\left(-8.09884…\\right)^{3}+3\\left(-8.09884…\\right)^{2}+2\\left(-8.09884…\\right)+1=-1943.27695…$$", "$$v_{12}=-8.09884…-\\frac{3829.49182…}{-1943.27695…}=-6.12820…$$" ], "result": "v_{12}=-6.12820…" }, { "type": "step", "primary": "$$Δv_{12}=\\left|-6.12820…-\\left(-8.09884…\\right)\\right|=1.97063…$$", "result": "Δv_{12}=1.97063…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{13}=-4.64785…{\\quad:\\quad}Δv_{13}=1.48034…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{12}\\right)=\\left(-6.12820…\\right)^{4}+\\left(-6.12820…\\right)^{3}+\\left(-6.12820…\\right)^{2}+\\left(-6.12820…\\right)+1=1212.65418…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{12}\\right)=4\\left(-6.12820…\\right)^{3}+3\\left(-6.12820…\\right)^{2}+2\\left(-6.12820…\\right)+1=-819.16882…$$", "$$v_{13}=-6.12820…-\\frac{1212.65418…}{-819.16882…}=-4.64785…$$" ], "result": "v_{13}=-4.64785…" }, { "type": "step", "primary": "$$Δv_{13}=\\left|-4.64785…-\\left(-6.12820…\\right)\\right|=1.48034…$$", "result": "Δv_{13}=1.48034…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{14}=-3.53453…{\\quad:\\quad}Δv_{14}=1.11332…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{13}\\right)=\\left(-4.64785…\\right)^{4}+\\left(-4.64785…\\right)^{3}+\\left(-4.64785…\\right)^{2}+\\left(-4.64785…\\right)+1=384.22115…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{13}\\right)=4\\left(-4.64785…\\right)^{3}+3\\left(-4.64785…\\right)^{2}+2\\left(-4.64785…\\right)+1=-345.11129…$$", "$$v_{14}=-4.64785…-\\frac{384.22115…}{-345.11129…}=-3.53453…$$" ], "result": "v_{14}=-3.53453…" }, { "type": "step", "primary": "$$Δv_{14}=\\left|-3.53453…-\\left(-4.64785…\\right)\\right|=1.11332…$$", "result": "Δv_{14}=1.11332…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{15}=-2.69527…{\\quad:\\quad}Δv_{15}=0.83926…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{14}\\right)=\\left(-3.53453…\\right)^{4}+\\left(-3.53453…\\right)^{3}+\\left(-3.53453…\\right)^{2}+\\left(-3.53453…\\right)+1=121.87505…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{14}\\right)=4\\left(-3.53453…\\right)^{3}+3\\left(-3.53453…\\right)^{2}+2\\left(-3.53453…\\right)+1=-145.21705…$$", "$$v_{15}=-3.53453…-\\frac{121.87505…}{-145.21705…}=-2.69527…$$" ], "result": "v_{15}=-2.69527…" }, { "type": "step", "primary": "$$Δv_{15}=\\left|-2.69527…-\\left(-3.53453…\\right)\\right|=0.83926…$$", "result": "Δv_{15}=0.83926…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{16}=-2.05895…{\\quad:\\quad}Δv_{16}=0.63632…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{15}\\right)=\\left(-2.69527…\\right)^{4}+\\left(-2.69527…\\right)^{3}+\\left(-2.69527…\\right)^{2}+\\left(-2.69527…\\right)+1=38.76232…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{15}\\right)=4\\left(-2.69527…\\right)^{3}+3\\left(-2.69527…\\right)^{2}+2\\left(-2.69527…\\right)+1=-60.91625…$$", "$$v_{16}=-2.69527…-\\frac{38.76232…}{-60.91625…}=-2.05895…$$" ], "result": "v_{16}=-2.05895…" }, { "type": "step", "primary": "$$Δv_{16}=\\left|-2.05895…-\\left(-2.69527…\\right)\\right|=0.63632…$$", "result": "Δv_{16}=0.63632…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{17}=-1.56818…{\\quad:\\quad}Δv_{17}=0.49077…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{16}\\right)=\\left(-2.05895…\\right)^{4}+\\left(-2.05895…\\right)^{3}+\\left(-2.05895…\\right)^{2}+\\left(-2.05895…\\right)+1=12.42335…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{16}\\right)=4\\left(-2.05895…\\right)^{3}+3\\left(-2.05895…\\right)^{2}+2\\left(-2.05895…\\right)+1=-25.31395…$$", "$$v_{17}=-2.05895…-\\frac{12.42335…}{-25.31395…}=-1.56818…$$" ], "result": "v_{17}=-1.56818…" }, { "type": "step", "primary": "$$Δv_{17}=\\left|-1.56818…-\\left(-2.05895…\\right)\\right|=0.49077…$$", "result": "Δv_{17}=0.49077…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{18}=-1.16736…{\\quad:\\quad}Δv_{18}=0.40081…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{17}\\right)=\\left(-1.56818…\\right)^{4}+\\left(-1.56818…\\right)^{3}+\\left(-1.56818…\\right)^{2}+\\left(-1.56818…\\right)+1=4.08216…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{17}\\right)=4\\left(-1.56818…\\right)^{3}+3\\left(-1.56818…\\right)^{2}+2\\left(-1.56818…\\right)+1=-10.18459…$$", "$$v_{18}=-1.56818…-\\frac{4.08216…}{-10.18459…}=-1.16736…$$" ], "result": "v_{18}=-1.16736…" }, { "type": "step", "primary": "$$Δv_{18}=\\left|-1.16736…-\\left(-1.56818…\\right)\\right|=0.40081…$$", "result": "Δv_{18}=0.40081…" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$v_{19}=-0.76245…{\\quad:\\quad}Δv_{19}=0.40490…$$", "steps": [ { "type": "step", "primary": "$$f\\left(v_{18}\\right)=\\left(-1.16736…\\right)^{4}+\\left(-1.16736…\\right)^{3}+\\left(-1.16736…\\right)^{2}+\\left(-1.16736…\\right)+1=1.46161…$$", "secondary": [ "$$f^{^{\\prime}}\\left(v_{18}\\right)=4\\left(-1.16736…\\right)^{3}+3\\left(-1.16736…\\right)^{2}+2\\left(-1.16736…\\right)+1=-3.60974…$$", "$$v_{19}=-1.16736…-\\frac{1.46161…}{-3.60974…}=-0.76245…$$" ], "result": "v_{19}=-0.76245…" }, { "type": "step", "primary": "$$Δv_{19}=\\left|-0.76245…-\\left(-1.16736…\\right)\\right|=0.40490…$$", "result": "Δv_{19}=0.40490…" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "Cannot find solution" } ], "meta": { "interimType": "Newton Raphson Find Real Solution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjALZJIPfVQGIeIQV5JvpOGUONOjZUnYwG7tfzq2B8KVS0QWTx3vXy+3rJkf5QRUUlWanAX5CivPcsI6uQ4LY8E+VPFsSrYxiR7WqQy7NlDscd/JUJRroD486Rh7zIYdtR/AV95ouxYbhT9hmx3c2nAq1sD7NfhsPe7eDHrmjY0mE8ROzzRiV/WzHpTRU/mmLvYBpyLFQQEQT3LVSYSE7hebYXTBRBHUGnRgUi8xd6rqD0BbPSc7ROvw/06D3ESKqojNHfOwYnV7aMLeoXuGfXFjvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "primary": "The solution is", "result": "\\mathrm{No\\:Solution\\:for}\\:v\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are", "result": "v=0,\\:v=-1,\\:v=1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "v=0,\\:v=-1,\\:v=1" }, { "type": "step", "primary": "Substitute back $$v=u^{2},\\:$$solve for $$u$$" }, { "type": "interim", "title": "Solve $$u^{2}=0:{\\quad}u=0$$", "input": "u^{2}=0", "steps": [ { "type": "step", "primary": "Apply rule $$x^n=0\\quad\\Rightarrow\\quad\\:x=0$$" }, { "type": "step", "result": "u=0" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$u^{2}=-1:{\\quad}$$No Solution for $$u\\in\\mathbb{R}$$", "input": "u^{2}=-1", "steps": [ { "type": "step", "primary": "$$x^{2}$$ cannot be negative for $$x\\in\\mathbb{R}$$", "result": "\\mathrm{No\\:Solution\\:for}\\:u\\in\\mathbb{R}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "interim", "title": "Solve $$u^{2}=1:{\\quad}u=1,\\:u=-1$$", "input": "u^{2}=1", "steps": [ { "type": "step", "primary": "For $$x^{2}=f\\left(a\\right)$$ the solutions are $$x=\\sqrt{f\\left(a\\right)},\\:\\:-\\sqrt{f\\left(a\\right)}$$" }, { "type": "step", "result": "u=\\sqrt{1},\\:u=-\\sqrt{1}" }, { "type": "interim", "title": "$$\\sqrt{1}=1$$", "input": "\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KfzlHGGU7KN8vfEO0eL8NN13jtrSFDx+UNsawjlOjV3ZuCguaNudj5qbY1K8A+fScubCnYZOJ5L8/2gsdymw1PSOscTE6qsKVI9GkIdY/eI=" } }, { "type": "interim", "title": "$$-\\sqrt{1}=-1$$", "input": "-\\sqrt{1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\sqrt{1}=1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7kWDE1Jsjy5jGSP2mctwcnCAn9lkDfZkicUGkO3EF+IpIQKToZa7Vmz9RWrIHzooCMHIu6EZfZrJ7HpyNTqg74lPlyk515FWfACaTxs0eUEM=" } }, { "type": "step", "result": "u=1,\\:u=-1" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solutions are" }, { "type": "step", "result": "u=0,\\:u=1,\\: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,\\:\\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": "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": "step", "primary": "Combine all the solutions", "result": "x=2πn,\\:x=π+2πn,\\:x=\\frac{π}{2}+2πn,\\: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^{22}(x)-\\sin^{2}(x)" }, "showViewLarger": true } }, "meta": { "showVerify": true } }