polar (-(9sqrt(3))/2 , 9/2)

{ "query": { "display": "cartesian to polar $$\\left(-\\frac{9\\sqrt{3}}{2},\\:\\frac{9}{2}\\right)$$", "symbolab_question": "POLAR#polar (-\\frac{9\\sqrt{3}}{2},\\frac{9}{2})" }, "solution": { "level": "PERFORMED", "subject": "Pre Calculus", "topic": "Polar Coordinates", "subTopic": "Polar", "default": "(9,-\\frac{π}{6}+π)" }, "steps": { "type": "interim", "title": "Convert $$\\left(-\\frac{9\\sqrt{3}}{2},\\:\\frac{9}{2}\\right)\\:$$to polar coordinates:$${\\quad}\\left(9,\\:-\\frac{π}{6}+π\\right)$$", "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": [ "$$x=-\\frac{9\\sqrt{3}}{2}$$", "$$y=\\frac{9}{2}$$" ] }, { "type": "step", "primary": "$$r=\\sqrt{x^2+y^2}$$", "result": "r=\\sqrt{\\left(-\\frac{9\\sqrt{3}}{2}\\right)^{2}+\\left(\\frac{9}{2}\\right)^{2}}" }, { "type": "interim", "title": "$$\\sqrt{\\left(-\\frac{9\\sqrt{3}}{2}\\right)^{2}+\\left(\\frac{9}{2}\\right)^{2}}=9$$", "input": "\\sqrt{\\left(-\\frac{9\\sqrt{3}}{2}\\right)^{2}+\\left(\\frac{9}{2}\\right)^{2}}", "steps": [ { "type": "interim", "title": "$$\\left(-\\frac{9\\sqrt{3}}{2}\\right)^{2}=\\frac{9^{2}\\cdot\\:3}{2^{2}}$$", "input": "\\left(-\\frac{9\\sqrt{3}}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-\\frac{9\\sqrt{3}}{2}\\right)^{2}=\\left(\\frac{9\\sqrt{3}}{2}\\right)^{2}$$" ], "result": "=\\left(\\frac{9\\sqrt{3}}{2}\\right)^{2}" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{\\left(9\\sqrt{3}\\right)^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$", "secondary": [ "$$\\left(9\\sqrt{3}\\right)^{2}=9^{2}\\left(\\sqrt{3}\\right)^{2}$$" ], "result": "=\\frac{9^{2}\\left(\\sqrt{3}\\right)^{2}}{2^{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": "=\\frac{9^{2}\\cdot\\:3}{2^{2}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Flp1pXKdlKMehaXWmt6U5llCzFZJV2W1So+1bM1nFpNV00rpv8+ZC6TM10tVCSHs0xDS+Y5aj0hl+F6LvDaAllD6NzCQnLtAW0DK3IeDCIuxc9UXHTUM7gxPijNXhuHZhQIXZbcagr11EwtLet1wvyMUplSqBJbUQog5wmtp2ZpMLtr+0cWa+2+XpLBchQxUTlJoneLjveZWr103b+MFDbCI2sSeA74029n2yo277ZU=" } }, { "type": "interim", "title": "$$\\left(\\frac{9}{2}\\right)^{2}=\\frac{9^{2}}{2^{2}}$$", "input": "\\left(\\frac{9}{2}\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(\\frac{a}{b}\\right)^{c}=\\frac{a^{c}}{b^{c}}$$", "result": "=\\frac{9^{2}}{2^{2}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78IrnuU8SCQIvZ9I2Lewbbo5IpdliG1E4K4EtDGLN9yvMwViaLUXkeD+JukROhWdjRJsWpAhoxuOoc1vOPwFo3aBtZyu+LaYvasa3tFjSwOF2Lb+s5nniFFgiZIXvgEQSaDa8rlf7v82lJeVK6kuZidjYXtkQysHlZUFTXnuM24M=" } }, { "type": "step", "result": "=\\sqrt{\\frac{9^{2}\\cdot\\:3}{2^{2}}+\\frac{9^{2}}{2^{2}}}" }, { "type": "interim", "title": "Combine the fractions $$\\frac{9^{2}\\cdot\\:3}{2^{2}}+\\frac{9^{2}}{2^{2}}:{\\quad}9^{2}$$", "result": "=\\sqrt{9^{2}}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{9^{2}\\cdot\\:3+9^{2}}{2^{2}}" }, { "type": "step", "primary": "Add similar elements: $$9^{2}\\cdot\\:3+9^{2}=9^{2}\\cdot\\:4$$", "result": "=\\frac{9^{2}\\cdot\\:4}{2^{2}}" }, { "type": "interim", "title": "Factor $$4:{\\quad}2^{2}$$", "steps": [ { "type": "step", "primary": "Factor $$4=2^{2}$$" } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{2^{2}\\cdot\\:9^{2}}{2^{2}}" }, { "type": "step", "primary": "Cancel the common factor: $$2^{2}$$", "result": "=9^{2}" } ], "meta": { "interimType": "LCD Top Title 1Eq" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a,\\:\\quad$$ assuming $$a\\ge0$$", "result": "=9", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Tn2zlUYUOt6wabPgavUPtBttQZaejSiwIWFmj9/NFUdNi8SYVYZf4sOrxKyGPNNtdau+gvDms31NhXKLyOuU0AOfOVs9mPIqDLV5QIWwt3l3De9kxFk/aYMHpYHnmkC2/d1Jf06TcLTuSuh3Tva+67ql6hWBkRwIJdIct7qdHq9QFcORX6gP6TA55DCsKvr56vRUo5Z6lUWQcFFe0+4mYLCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "r=9" }, { "type": "step", "primary": "$$\\theta=\\arctan\\left(\\frac{y}{x}\\right)$$", "result": "θ=\\arctan\\left(\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}\\right)" }, { "type": "interim", "title": "Adjust $$\\theta$$ based on the quadrant of the point $$\\left(-\\frac{9\\sqrt{3}}{2},\\:\\frac{9}{2}\\right)$$", "result": "θ=\\arctan\\left(\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}\\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": [ "$$\\left(-\\frac{9\\sqrt{3}}{2},\\:\\frac{9}{2}\\right)\\:$$is in quadrant II" ] }, { "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{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}\\right)+π" } ], "meta": { "interimType": "Cartesian To Polar Adjust Theta 1Eq" } }, { "type": "interim", "title": "$$\\arctan\\left(\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}\\right)+π=-\\frac{π}{6}+π$$", "input": "\\arctan\\left(\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}\\right)+π", "steps": [ { "type": "interim", "title": "$$\\arctan\\left(\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}\\right)=-\\frac{π}{6}$$", "input": "\\arctan\\left(\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}=-\\frac{1}{\\sqrt{3}}$$", "input": "\\frac{\\frac{9}{2}}{-\\frac{9\\sqrt{3}}{2}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{\\frac{9}{2}}{\\frac{9\\sqrt{3}}{2}}" }, { "type": "step", "primary": "Divide fractions: $$\\frac{\\frac{a}{b}}{\\frac{c}{d}}=\\frac{a\\cdot\\:d}{b\\cdot\\:c}$$", "result": "=-\\frac{9\\cdot\\:2}{2\\cdot\\:9\\sqrt{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$9$$", "result": "=-\\frac{2}{2\\sqrt{3}}" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{1}{\\sqrt{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7tiTmKkmeEqHWILbBb88ajcSQSmFSt2ksruwxKkQwZo7Eikj866oQtd+9K/T9kTc/VdNK6b/PmQukzNdLVQkh7IYBZDEshoZHCbn1arDHqvREdDvphBur04TGefg6xqZJP8vQyhiD4JSfqjIvcQ7tiuKshYb/aMgpUXqqV2Z6Pa5c+2M6IP1TqarB3oS0R8CbHOrk4mePk7KAVl2KjxTOxXW0gmWYFGZuqMCgRprVvxcWRUPDnIQXYMTDiaxDMpN7" } }, { "type": "step", "result": "=\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)" }, { "type": "interim", "title": "$$\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)=\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)$$", "input": "\\arctan\\left(-\\frac{1}{\\sqrt{3}}\\right)", "steps": [ { "type": "interim", "title": "$$\\frac{1}{\\sqrt{3}}=\\frac{\\sqrt{3}}{3}$$", "input": "\\frac{1}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Multiply by the conjugate $$\\frac{\\sqrt{3}}{\\sqrt{3}}$$", "result": "=\\frac{1\\cdot\\:\\sqrt{3}}{\\sqrt{3}\\sqrt{3}}", "meta": { "title": { "extension": "To rationalize the denominator, multiply numerator and denominator by the conjugate of the radical $$\\sqrt{3}$$" } } }, { "type": "step", "primary": "$$1\\cdot\\:\\sqrt{3}=\\sqrt{3}$$" }, { "type": "interim", "title": "$$\\sqrt{3}\\sqrt{3}=3$$", "input": "\\sqrt{3}\\sqrt{3}", "result": "=\\frac{\\sqrt{3}}{3}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$", "secondary": [ "$$\\sqrt{3}\\sqrt{3}=3$$" ], "result": "=3", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KTpmXttz9tzhFROsvdfdpl9t0rGzXfqIeCX9GJLyBFHMwViaLUXkeD+JukROhWdjqqRb86vK1jAPxwJmTBc7jxuwBfr4jqpPqXmFXXvxZ1uqe0B1E2GM2M0eTcXoyACf" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78B+yE0cCl8hT7YErNdnZGYH5WPs65D84evxXUHATCeerju+5Z51e/ZZSD3gRHwjBnvjDY21b5XBQ44AG3rKeoOdIubRDLxVLz5gHGSWn9gz//NvXaLneG2moeSe54R9ofl+2Q1REnsr4pFHHZPAlAOqrxevU0S7BviPFPoEcZw6wiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7EO1WRSe7TxupdIW8DpQ6akR0O+mEG6vThMZ5+DrGpknTLx8mOdHYVzxX643JqKFIQslTDKxOR/6J+ZOGvUcauphMhI1dYkQcBy/Dfx/H0IxCNWox8otddtwDvOqnWCVPuK/CPOCJfkN3rVLMAJKBQwniDfxUdAs4y15bjJftwC5syiQ1IhsnfU83dsVfC+ZGfbhFjKUqusZ1g6Yd3VRGiE3WhGuA6u9cCyUkTGrB//4=" } }, { "type": "step", "result": "=\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)" }, { "type": "step", "primary": "Use the following property: $$\\arctan\\left(-x\\right)=-\\arctan\\left(x\\right)$$", "secondary": [ "$$\\arctan\\left(-\\frac{\\sqrt{3}}{3}\\right)=-\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)$$" ], "result": "=-\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)" }, { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)=\\frac{π}{6}$$", "input": "\\arctan\\left(\\frac{\\sqrt{3}}{3}\\right)", "steps": [ { "type": "step", "primary": "$$\\begin{array}{|c|c|c|}\\hline x&\\arctan(x)&\\arctan(x)\\\\\\hline 0&0&0^{\\circ}\\\\\\hline \\frac{\\sqrt{3}}{3}&\\frac{\\pi}{6}&30^{\\circ}\\\\\\hline 1&\\frac{\\pi}{4}&45^{\\circ}\\\\\\hline \\sqrt{3}&\\frac{\\pi}{3}&60^{\\circ}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=\\frac{π}{6}" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=-\\frac{π}{6}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iGvFWxodO1NUsa8/NPZQs7BXdanoOq8NueHe605tUWpnLvWYUX1v4TcjVreYjjEgbBmAlTNXuPi//ufFZJsjQXWD310L1+P2yDQQfMEhENEQnwAwcA1rV7fIxYB3Gnm0AYgvLkGJ3T/kTR+/qIhEJ4NJJ6BQ0z5tdkEckLJlqak4BIvlQW+Pl+FjMap6p8H0Fg/WfmX2M9LkArUCIhhXe70Y6isfoMyGu1cz+ZxUHb2Qe+SNKDnjTwuozJW1bVni" } }, { "type": "step", "result": "=-\\frac{π}{6}+π" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iGvFWxodO1NUsa8/NPZQs7BXdanoOq8NueHe605tUWpnLvWYUX1v4TcjVreYjjEgObB7NjuQSMFNRwU5HT33O913jtrSFDx+UNsawjlOjV2xpuaZH1kFD7Iu9AlpaofQHA7lnE0yR79KmkWxQ7S+h5A2wMwkAjw4bn/47I8ARYoL747jO5GYuttbAUKi0lDSxJBKYVK3aSyu7DEqRDBmjgcTEZvAimQjsHFXreMA7LROH7U08jE/njm199dwAdoKwlJVPOY8/XJgNq9r6BtAPQ==" } }, { "type": "step", "result": "θ=-\\frac{π}{6}+π" }, { "type": "step", "primary": "The polar coordinates of $$\\left(-\\frac{9\\sqrt{3}}{2},\\:\\frac{9}{2}\\right)$$", "result": "\\left(9,\\:-\\frac{π}{6}+π\\right)" } ] } }