{
"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}+π)"
},
"steps": {
"type": "interim",
"title": "Convert $$\\left(-2\\sqrt{3},\\:-2\\right)\\:$$to polar coordinates:$${\\quad}\\left(4,\\:\\frac{π}{6}+π\\right)$$",
"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}+\\left(-2\\right)^{2}}"
},
{
"type": "interim",
"title": "$$\\sqrt{\\left(-2\\sqrt{3}\\right)^{2}+\\left(-2\\right)^{2}}=4$$",
"input": "\\sqrt{\\left(-2\\sqrt{3}\\right)^{2}+\\left(-2\\right)^{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: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even",
"secondary": [
"$$\\left(-2\\sqrt{3}\\right)^{2}=\\left(2\\sqrt{3}\\right)^{2}$$"
],
"result": "=\\left(2\\sqrt{3}\\right)^{2}"
},
{
"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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79jevKP+ZSk/HvopBKHFRKydVBn2NNCFZqg4ZoVh6UwqjkVi15I8rBefLi4Iyt2wryRx8VhzYaNA4GqhP6DSGa/pdN8JYpy3fJ0d6WFWhpqWcftg6wCsBAuVxsDQAkYl4MdGh0b4sYpbe5HFlR6NPJA=="
}
},
{
"type": "interim",
"title": "$$\\left(-2\\right)^{2}=2^{2}$$",
"input": "\\left(-2\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even",
"secondary": [
"$$\\left(-2\\right)^{2}=2^{2}$$"
],
"result": "=2^{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sNuhSfBo+/I8oqMceBlhcs0ag8T1MwTer44+aCS/ZFBDeoKWfP4f0hW8hp+DjlqkWG48kfKlXwh1JXHkPaftrOeZImDuB9kLWbJJECF6RjY="
}
},
{
"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": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7haVRPAP+HIuvPdItN6zbj5g4au+tH5EMmPqbAzxuSzwpFpxf/FsVs2fet+df9/JgzMFYmi1F5Hg/ibpEToVnYz8s1oAoBG9v5706TVT42P6tm6R5QpOgKHJ64KDcz7O4mDhq760fkQyY+psDPG5LPCkJyLuOCyFfZCUBtrsLABE="
}
},
{
"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": "Adjust $$\\theta$$ based on the quadrant of the point $$\\left(-2\\sqrt{3},\\:-2\\right)$$",
"result": "θ=\\arctan\\left(\\frac{-2}{-2\\sqrt{3}}\\right)+π",
"steps": [
{
"type": "definition",
"title": "Point location",
"text": "If x>0, y>0, then the point is in quadrant I<br/>If x<0, y>0, then the point is in quadrant II<br/>If x<0, and y<0, then the point is in quadrant III<br/>If x>0, and y<0, then the point is in quadrant IV",
"secondary": [
"$$\\left(-2\\sqrt{3},\\:-2\\right)\\:$$is in quadrant III"
]
},
{
"type": "step",
"primary": "If in quadrant II or III, add $$\\pi$$ to $$\\theta$$<br/>If in quadrant IV, add $$2\\pi$$ to $$\\theta$$",
"result": "θ=\\arctan\\left(\\frac{-2}{-2\\sqrt{3}}\\right)+π"
}
],
"meta": {
"interimType": "Cartesian To Polar Adjust Theta 1Eq"
}
},
{
"type": "interim",
"title": "$$\\arctan\\left(\\frac{-2}{-2\\sqrt{3}}\\right)+π=\\frac{π}{6}+π$$",
"input": "\\arctan\\left(\\frac{-2}{-2\\sqrt{3}}\\right)+π",
"steps": [
{
"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": "interim",
"title": "$$\\frac{-2}{-2\\sqrt{3}}=\\frac{1}{\\sqrt{3}}$$",
"input": "\\frac{-2}{-2\\sqrt{3}}",
"steps": [
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$",
"result": "=\\frac{2}{2\\sqrt{3}}"
},
{
"type": "step",
"primary": "Divide the numbers: $$\\frac{2}{2}=1$$",
"result": "=\\frac{1}{\\sqrt{3}}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7FLXn50588Z3lYh7fA66Hc0qoKivTsDLBbpWfnfTqtRgDnzlbPZjyKgy1eUCFsLd5xXDODtFBCC8Uf836IcE9x1W5kakCgbdaDg0Dvq4S9qTNWyGcX6HZt1LGXH2QGa+LnevkT9T03BHga2KkaTm1dReBIDw621LE0nQ7i09r+QEBLYoznbQT9RDNBW3uT3ST"
}
},
{
"type": "step",
"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/NPZQs/ZQLoKb7+Mkpkr8DpJtI1zjJEJ7DGyZxnScUi0ELADjo5FYteSPKwXny4uCMrdsK998p0/KNUdTi/W27EuvVbC4r8I84Il+Q3etUswAkoFDEwFQkWMNNUWyLUsqXvMsRIWcdMltOg1NrRKLnod0N9IBJO1O9rFCBAvUuon865k0"
}
},
{
"type": "step",
"result": "=\\frac{π}{6}+π"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iGvFWxodO1NUsa8/NPZQs/ZQLoKb7+Mkpkr8DpJtI1z7doWj53z/ti3OewKSjJr+q47vuWedXv2WUg94ER8IwSEOZjx8J3zi9MgZHqT22UtwS+lHYVXegIqrUSJhY+Vig0knoFDTPm12QRyQsmWpqcs0J80zOogA5Nq1G+DKIchIGh8mPBAsB4r2j9dk6ly8QjUBcNiM9l2oxmLtfIzfyg=="
}
},
{
"type": "step",
"result": "θ=\\frac{π}{6}+π"
},
{
"type": "step",
"primary": "The polar coordinates of $$\\left(-2\\sqrt{3},\\:-2\\right)$$",
"result": "\\left(4,\\:\\frac{π}{6}+π\\right)"
}
]
}
}
Solution
cartesian to polar
Solution
Solution steps
Adjust based on the quadrant of the point
The polar coordinates of