{
"query": {
"display": "cartesiano a polar $$\\left(4\\sqrt{2},\\:4\\sqrt{2}\\right)$$",
"symbolab_question": "POLAR#polar (4\\sqrt{2},4\\sqrt{2})"
},
"solution": {
"level": "PERFORMED",
"subject": "Pre Calculus",
"topic": "Polar Coordinates",
"subTopic": "Polar",
"default": "(8,\\frac{π}{4})"
},
"steps": {
"type": "interim",
"title": "Convertir $$\\left(4\\sqrt{2},\\:4\\sqrt{2}\\right)\\:$$a coordenadas polares:$${\\quad}\\left(8,\\:\\frac{π}{4}\\right)$$",
"steps": [
{
"type": "definition",
"title": "Definición",
"text": "Para convertir coordenadas cartesianas $$\\left(x,\\:y\\right)$$ a coordenadas polares $$\\left(r,\\:\\theta\\right)$$ aplicar:<br/>$$r=\\sqrt{x^2+y^2}\\quad\\theta=\\arctan\\left(\\frac{y}{x}\\right)$$",
"secondary": [
"$$x=4\\sqrt{2}$$",
"$$y=4\\sqrt{2}$$"
]
},
{
"type": "step",
"primary": "$$r=\\sqrt{x^2+y^2}$$",
"result": "r=\\sqrt{\\left(4\\sqrt{2}\\right)^{2}+\\left(4\\sqrt{2}\\right)^{2}}"
},
{
"type": "interim",
"title": "$$\\sqrt{\\left(4\\sqrt{2}\\right)^{2}+\\left(4\\sqrt{2}\\right)^{2}}=8$$",
"input": "\\sqrt{\\left(4\\sqrt{2}\\right)^{2}+\\left(4\\sqrt{2}\\right)^{2}}",
"steps": [
{
"type": "step",
"primary": "Sumar elementos similares: $$\\left(4\\sqrt{2}\\right)^{2}+\\left(4\\sqrt{2}\\right)^{2}=2\\left(4\\sqrt{2}\\right)^{2}$$",
"result": "=\\sqrt{2\\left(4\\sqrt{2}\\right)^{2}}"
},
{
"type": "interim",
"title": "$$\\left(4\\sqrt{2}\\right)^{2}=4^{2}\\cdot\\:2$$",
"input": "\\left(4\\sqrt{2}\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=4^{2}\\left(\\sqrt{2}\\right)^{2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\left(\\sqrt{2}\\right)^{2}:{\\quad}2$$",
"steps": [
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt{a}=a^{\\frac{1}{2}}$$",
"result": "=\\left(2^{\\frac{1}{2}}\\right)^{2}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=2^{\\frac{1}{2}\\cdot\\:2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\frac{1}{2}\\cdot\\:2=1$$",
"input": "\\frac{1}{2}\\cdot\\:2",
"result": "=2",
"steps": [
{
"type": "step",
"primary": "Multiplicar fracciones: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:2}{2}"
},
{
"type": "step",
"primary": "Eliminar los terminos comunes: $$2$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/"
}
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=4^{2}\\cdot\\:2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7OsiCWAZacTD6LiZfJPFrYFnyYRz18HvB+rp63mPitc88LIDuLGMJ18Nh77Jpzso65vgxwMbwXxK4lQ5R1CLZmCG7m9WnfybyYkbXo1FDZwUr/7bEkkhk6oKuilPiOgh+SH3X0cd0z2mxOSnU6ivEHQ=="
}
},
{
"type": "step",
"result": "=\\sqrt{4^{2}\\cdot\\:2\\cdot\\:2}"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$2\\cdot\\:2=4$$",
"result": "=\\sqrt{4^{2}\\cdot\\:4}"
},
{
"type": "interim",
"title": "Simplificar $$4\\cdot\\:4^{2}:{\\quad}4^{3}$$",
"input": "4\\cdot\\:4^{2}",
"result": "=\\sqrt{4^{3}}",
"steps": [
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$4\\cdot\\:4^{2}=\\:4^{1+2}$$"
],
"result": "=4^{1+2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Sumar: $$1+2=3$$",
"result": "=4^{3}"
}
],
"meta": {
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$a^{b+c}=a^{b}a^{c}$$",
"result": "=\\sqrt{4^{2}\\cdot\\:4^{1}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt{a\\cdot{b}}=\\sqrt{a}\\sqrt{b}$$",
"result": "=\\sqrt{4^{2}}\\sqrt{4^{1}}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Aplicar la siguiente propiedad de los radicales: $$\\sqrt[n]{a^n}=a,\\:\\quad$$asumiendo que $$a\\ge0$$",
"secondary": [
"$$\\sqrt{4^{2}}=4$$"
],
"result": "=4\\sqrt{4^{1}}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Aplicar la regla $$a^{1}=a$$",
"secondary": [
"$$4^{1}=4$$"
],
"result": "=4\\sqrt{4}"
},
{
"type": "interim",
"title": "$$\\sqrt{4}=2$$",
"input": "\\sqrt{4}",
"steps": [
{
"type": "step",
"primary": "Descomponer el número en factores primos: $$4=2^{2}$$",
"result": "=\\sqrt{2^{2}}"
},
{
"type": "step",
"primary": "Aplicar las leyes de los exponentes: $$\\sqrt[n]{a^n}=a$$",
"secondary": [
"$$\\sqrt{2^{2}}=2$$"
],
"result": "=2",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RnzZTJ4FPnsHVA8/0U5Nl913jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WcubCnYZOJ5L8/2gsdymw1DH70PdnXJfHf+8MsVWHq0c="
}
},
{
"type": "step",
"result": "=4\\cdot\\:2"
},
{
"type": "step",
"primary": "Multiplicar los numeros: $$4\\cdot\\:2=8$$",
"result": "=8"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s73ojoTLhToL5oF7/tYABiBEatdaGlQSce6CSM0qT8d2c1eyJQh/MI5LZeYs7xEY3D3XeO2tIUPH5Q2xrCOU6NXYDqVf+OuFFu3cWSYbdrcrhy5sKdhk4nkvz/aCx3KbDUXP9yYxNMCWfSK5w4vgFn1Trn/PLRM68B4oHFvwam7Ll/muf9+bI8PxuLvv29nu7h"
}
},
{
"type": "step",
"result": "r=8"
},
{
"type": "step",
"primary": "$$\\theta=\\arctan\\left(\\frac{y}{x}\\right)$$",
"result": "θ=\\arctan\\left(\\frac{4\\sqrt{2}}{4\\sqrt{2}}\\right)"
},
{
"type": "interim",
"title": "$$\\arctan\\left(\\frac{4\\sqrt{2}}{4\\sqrt{2}}\\right)=\\frac{π}{4}$$",
"input": "\\arctan\\left(\\frac{4\\sqrt{2}}{4\\sqrt{2}}\\right)",
"steps": [
{
"type": "step",
"primary": "Aplicar la regla $$\\frac{a}{a}=1$$",
"result": "=\\arctan\\left(1\\right)"
},
{
"type": "step",
"primary": "Utilizar la siguiente identidad trivial:$${\\quad}\\arctan\\left(1\\right)=\\frac{π}{4}$$",
"secondary": [
"$$\\begin{array}{|c|c|c|}\\hline x&\\arctan(x)&\\arctan(x)\\\\\\hline 0&0&0^{\\circ}\\\\\\hline \\frac{\\sqrt{3}}{3}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline 1&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline \\sqrt{3}&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline \\end{array}$$"
],
"result": "=\\frac{π}{4}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iGvFWxodO1NUsa8/NPZQs4eIZ3MGMvoO4euueoRKLO6v3EvSw2yOOAryCKkNEOHHzRqDxPUzBN6vjj5oJL9kUFe8Gp704IyqpyA/1DKuuhtTtU9LJxTj6ily/kujMVZpe44hXR69ElaJv5InEb8DK6fRFerz/W3iWk8hntWNGJ49bk6iGJ9ZwFAGB4jwVZVUX5WUC1bh7kZtdxXEHdfaJA=="
}
},
{
"type": "step",
"result": "θ=\\frac{π}{4}"
},
{
"type": "step",
"primary": "Las coordenadas polares de $$\\left(4\\sqrt{2},\\:4\\sqrt{2}\\right)$$",
"result": "\\left(8,\\:\\frac{π}{4}\\right)"
}
]
}
}
Solución
cartesiano a polar
Solución
Pasos de solución
Las coordenadas polares de