{
"query": {
"display": "cartesian to polar $$\\left(2,\\:2\\right)$$",
"symbolab_question": "POLAR#polar (2,2)"
},
"solution": {
"level": "PERFORMED",
"subject": "Pre Calculus",
"topic": "Polar Coordinates",
"subTopic": "Polar",
"default": "(2\\sqrt{2},\\frac{π}{4})"
},
"steps": {
"type": "interim",
"title": "Convert $$\\left(2,\\:2\\right)\\:$$to polar coordinates:$${\\quad}\\left(2\\sqrt{2},\\:\\frac{π}{4}\\right)$$",
"steps": [
{
"type": "definition",
"title": "Definition",
"text": "To convert Cartesian coordinates $$\\left(x,\\:y\\right)\\:$$to Polar coordinates $$\\left(r,\\:\\theta\\right)\\:$$apply:<br/>$$r=\\sqrt{x^2+y^2}\\quad\\theta=\\arctan\\left(\\frac{y}{x}\\right)$$",
"secondary": [
"$$x=2$$",
"$$y=2$$"
]
},
{
"type": "step",
"primary": "$$r=\\sqrt{x^2+y^2}$$",
"result": "r=\\sqrt{2^{2}+2^{2}}"
},
{
"type": "interim",
"title": "$$\\sqrt{2^{2}+2^{2}}=2\\sqrt{2}$$",
"input": "\\sqrt{2^{2}+2^{2}}",
"steps": [
{
"type": "interim",
"title": "Simplify $$2^{2}+2^{2}:{\\quad}2^{3}$$",
"input": "2^{2}+2^{2}",
"result": "=\\sqrt{2^{3}}",
"steps": [
{
"type": "step",
"primary": "Add similar elements: $$2^{2}+2^{2}=2^{2}\\cdot\\:2$$",
"result": "=2^{2}\\cdot\\:2"
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$2\\cdot\\:2^{2}=\\:2^{1+2}$$"
],
"result": "=2^{1+2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$1+2=3$$",
"result": "=2^{3}"
}
],
"meta": {
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$",
"result": "=\\sqrt{2^{2}\\cdot\\:2^{1}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{a\\cdot{b}}=\\sqrt{a}\\sqrt{b}$$",
"result": "=\\sqrt{2^{2}}\\sqrt{2^{1}}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$",
"secondary": [
"$$\\sqrt{2^{2}}=2$$"
],
"result": "=2\\sqrt{2^{1}}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply rule $$a^{1}=a$$",
"secondary": [
"$$2^{1}=2$$"
],
"result": "=2\\sqrt{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tL5k06ZlxVcL+GVDJ+Kh3l1iJRJdZU8Snq+IOvMxYQFwkKGJWEPFPk38sdJMsyPI9luHUZAWv4RSdMvOgJQIrYY+spUsB4yZJkDyh9+Pu3/uuyFHPQmdXE3hYjHq+tbv1e48pqXVwLw5QfZ/28IfyA=="
}
},
{
"type": "step",
"result": "r=2\\sqrt{2}"
},
{
"type": "step",
"primary": "$$\\theta=\\arctan\\left(\\frac{y}{x}\\right)$$",
"result": "θ=\\arctan\\left(\\frac{2}{2}\\right)"
},
{
"type": "interim",
"title": "$$\\arctan\\left(\\frac{2}{2}\\right)=\\frac{π}{4}$$",
"input": "\\arctan\\left(\\frac{2}{2}\\right)",
"steps": [
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{a}=1$$",
"result": "=\\arctan\\left(1\\right)"
},
{
"type": "step",
"primary": "Use the following trivial identity:$${\\quad}\\arctan\\left(1\\right)=\\frac{π}{4}$$",
"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/NPZQs1+FXwyAMaDviYTsyK1RfFh1g99dC9fj9sg0EHzBIRDRlkAKapmtBOxPQtJEBZtsA3E2Z5v/GFq2v9cvwjSfE0ymITG4iqRh0gJvf9AVTdkoNfk6gp+2wolB8q15NP5KQtiVPjD2sg2GG++2CSgE0w8="
}
},
{
"type": "step",
"result": "θ=\\frac{π}{4}"
},
{
"type": "step",
"primary": "The polar coordinates of $$\\left(2,\\:2\\right)$$",
"result": "\\left(2\\sqrt{2},\\:\\frac{π}{4}\\right)"
}
]
}
}
Solution
cartesian to polar
Solution
Solution steps
The polar coordinates of