{
"query": {
"display": "cartesian to polar $$\\left(2\\sqrt{3},\\:2\\right)$$",
"symbolab_question": "POLAR#polar (2\\sqrt{3},2)"
},
"solution": {
"level": "PERFORMED",
"subject": "Pre Calculus",
"topic": "Polar Coordinates",
"subTopic": "Polar",
"default": "(4,\\frac{π}{6})"
},
"steps": {
"type": "interim",
"title": "Convert $$\\left(2\\sqrt{3},\\:2\\right)\\:$$to polar coordinates:$${\\quad}\\left(4,\\:\\frac{π}{6}\\right)$$",
"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": [
"$$x=2\\sqrt{3}$$",
"$$y=2$$"
]
},
{
"type": "step",
"primary": "$$r=\\sqrt{x^2+y^2}$$",
"result": "r=\\sqrt{\\left(2\\sqrt{3}\\right)^{2}+2^{2}}"
},
{
"type": "interim",
"title": "$$\\sqrt{\\left(2\\sqrt{3}\\right)^{2}+2^{2}}=4$$",
"input": "\\sqrt{\\left(2\\sqrt{3}\\right)^{2}+2^{2}}",
"steps": [
{
"type": "interim",
"title": "$$\\left(2\\sqrt{3}\\right)^{2}=2^{2}\\cdot\\:3$$",
"input": "\\left(2\\sqrt{3}\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=2^{2}\\left(\\sqrt{3}\\right)^{2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\left(\\sqrt{3}\\right)^{2}:{\\quad}3$$",
"steps": [
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{a}=a^{\\frac{1}{2}}$$",
"result": "=\\left(3^{\\frac{1}{2}}\\right)^{2}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=3^{\\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": "=3",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:2}{2}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$2$$",
"result": "=1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l3vTdf410Ywhq1vZ0kzF8e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBE9/03SOiEv+BIHutWLr6nUfz18ijmoplMAomfJM9x8W1GdKgiNs+PolKvTuWzYk/"
}
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=2^{2}\\cdot\\:3"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7365YmXBJU5aZ84stNhbIKVnyYRz18HvB+rp63mPitc+zsHBJV0oRhKqf7h8oBCgaU1Segk/l8hMTwwzmyvEPsbKAdlEWdnVS/2Z0ay5lcmnrMt0YSkP2lJNJOULcrhO3l0w8aVbUaQyDWZujTOqnuQ=="
}
},
{
"type": "step",
"result": "=\\sqrt{2^{2}\\cdot\\:3+2^{2}}"
},
{
"type": "step",
"primary": "Add similar elements: $$2^{2}\\cdot\\:3+2^{2}=2^{2}\\cdot\\:4$$",
"result": "=\\sqrt{2^{2}\\cdot\\:4}"
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b},\\:\\quad$$ assuming $$a\\ge0,\\:b\\ge0$$",
"result": "=\\sqrt{4}\\sqrt{2^{2}}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "interim",
"title": "$$\\sqrt{4}=2$$",
"input": "\\sqrt{4}",
"result": "=2\\sqrt{2^{2}}",
"steps": [
{
"type": "step",
"primary": "Factor the number: $$4=2^{2}$$",
"result": "=\\sqrt{2^{2}}"
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$",
"secondary": [
"$$\\sqrt{2^{2}}=2$$"
],
"result": "=2",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$",
"secondary": [
"$$\\sqrt{2^{2}}=2$$"
],
"result": "=2\\cdot\\:2",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:2=4$$",
"result": "=4"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CPVeoJiKqLByyAWbUZJC0kpj8gBYGzeJgQEFEhIVI4gAlilG71elit3w1IBbYN0Pdon1OhrviX+zjeY8LDixH6N6Hv6MoTMtvtU0IQwXdn8ssW9kOE6+fwYfBavYHcr4jhlPJTF4U/fpSpdQxLOBpiS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "r=4"
},
{
"type": "step",
"primary": "$$\\theta=\\arctan\\left(\\frac{y}{x}\\right)$$",
"result": "θ=\\arctan\\left(\\frac{2}{2\\sqrt{3}}\\right)"
},
{
"type": "interim",
"title": "$$\\arctan\\left(\\frac{2}{2\\sqrt{3}}\\right)=\\frac{π}{6}$$",
"input": "\\arctan\\left(\\frac{2}{2\\sqrt{3}}\\right)",
"steps": [
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{2}{2}=1$$",
"result": "=\\arctan\\left(\\frac{1}{\\sqrt{3}}\\right)"
},
{
"type": "step",
"primary": "Use the following trivial identity:$${\\quad}\\arctan\\left(\\frac{1}{\\sqrt{3}}\\right)=\\frac{π}{6}$$",
"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}{6}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iGvFWxodO1NUsa8/NPZQswEcDzhUXOvGAzKSjeoICr/TLx8mOdHYVzxX643JqKFIQslTDKxOR/6J+ZOGvUcaum2V9bcxPzf3wXediGjUckU6USmq8vZAJBXx71jJT3BY0+Q9JFLVbH5Gt90MFbYPWBToOTUVtH7BsqJ3B1OUhX+Qe+SNKDnjTwuozJW1bVni"
}
},
{
"type": "step",
"result": "θ=\\frac{π}{6}"
},
{
"type": "step",
"primary": "The polar coordinates of $$\\left(2\\sqrt{3},\\:2\\right)$$",
"result": "\\left(4,\\:\\frac{π}{6}\\right)"
}
]
}
}
Solution
cartesian to polar
Solution
Solution steps
The polar coordinates of