Consider the point [latex]\\left(1,\\sqrt{3}\\right)[\/latex] in rectangular coordinates. This same point can be expressed in polar form as:<\/p>\r\n\r\n
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- [latex]\\left(2,\\frac{\\pi }{3}\\right)[\/latex]<\/li>\r\n \t
- [latex]\\left(2,\\frac{7\\pi }{3}\\right)[\/latex]<\/li>\r\n \t
- [latex]\\left(-2,\\frac{4\\pi }{3}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n
\r\n Let's verify that [latex]\\left(-2,\\frac{4\\pi }{3}\\right)[\/latex] represents [latex]\\left(1,\\sqrt{3}\\right)[\/latex]:<\/p>\r\n\r\n
[latex]\\begin{array}{ccccccc}\\begin{array}{ccc}\\hfill x& =\\hfill & r\\cos\\theta \\hfill \\\\ & =\\hfill & -2\\cos\\left(\\frac{4\\pi }{3}\\right)\\hfill \\\\ & =\\hfill & -2\\left(-\\frac{1}{2}\\right)=1\\hfill \\end{array}\\hfill & & & \\text{and}\\hfill & & & \\begin{array}{ccc}\\hfill y& =\\hfill & r\\sin\\theta \\hfill \\\\ & =\\hfill & -2\\sin\\left(\\frac{4\\pi }{3}\\right)\\hfill \\\\ & =\\hfill & -2\\left(-\\frac{\\sqrt{3}}{2}\\right)=\\sqrt{3}.\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\r\n<\/section>\r\nThis confirms that negative values of [latex]r[\/latex] are allowed in polar coordinates. While every point has infinitely many polar representations, each point has only one representation in rectangular coordinates.<\/p>\r\n
The polar coordinate system has a clear visual interpretation. The value [latex]r[\/latex] represents the directed distance<\/strong> from the origin to the point, while [latex]\\theta[\/latex] measures the angle<\/strong> that the line segment makes with the positive [latex]x[\/latex]-axis.<\/p>\r\n
In the polar plane, the horizontal line extending right from the center is called the polar axis<\/strong> (equivalent to the positive [latex]x[\/latex]-axis). The center point is the pole<\/strong> or origin, corresponding to [latex]r=0[\/latex]. Concentric circles represent points at fixed distances from the pole. The equation [latex]r=1[\/latex] describes all points one unit from the pole, [latex]r=2[\/latex] describes points two units away, and so on. Line segments radiating from the pole correspond to fixed angles.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"576\"]
![\"A](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234809\/CNX_Calc_Figure_11_03_002.jpg\")
Figure 2. The polar coordinate system.[\/caption]\r\n\r\n\r\n How to: Plot Polar Points<\/strong>:<\/p>\r\n\r\n
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- Start with the angle [latex]\\theta[\/latex]. Measure counterclockwise from the polar axis if positive, clockwise if negative.<\/li>\r\n \t
- If [latex]r > 0[\/latex], move that distance along the terminal ray of the angle.<\/li>\r\n \t
- If [latex]r < 0[\/latex], move that distance along the ray opposite to the terminal ray.<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n
\r\n \r\nPlot each of the following points on the polar plane.<\/p>\r\n\r\n
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- [latex]\\left(2,\\frac{\\pi }{4}\\right)[\/latex]<\/li>\r\n \t
- [latex]\\left(-3,\\frac{2\\pi }{3}\\right)[\/latex]<\/li>\r\n \t
- [latex]\\left(4,\\frac{5\\pi }{4}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n[reveal-answer q=\"44558896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558896\"]\r\n\r\n
The three points are plotted in the following figure.<\/p>\r\n\r\n
<\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"] ![\"Three](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234812\/CNX_Calc_Figure_11_03_003.jpg\")
Figure 3. Three points plotted in the polar coordinate system.[\/caption]<\/figure>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video to see the worked solution to the example above.