{"id":224,"date":"2025-02-13T22:45:15","date_gmt":"2025-02-13T22:45:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-coordinates\/"},"modified":"2025-08-13T16:58:21","modified_gmt":"2025-08-13T16:58:21","slug":"polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polar-coordinates\/","title":{"raw":"Polar Coordinates: Learn It 2","rendered":"Polar Coordinates: Learn It 2"},"content":{"raw":"

Plotting Points Using Polar Coordinates<\/h2>\r\nWhen we think about plotting points in the plane, we usually think of rectangular coordinates<\/strong> [latex]\\left(x,y\\right)[\/latex] in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates<\/strong>, which are points labeled [latex]\\left(r,\\theta \\right)[\/latex] and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole<\/strong>, or the origin of the coordinate plane.\r\n\r\nThe polar grid<\/strong> is scaled as the unit circle with the positive x-<\/em>axis now viewed as the polar axis<\/strong> and the origin as the pole. The first coordinate [latex]r[\/latex] is the radius or length of the directed line segment from the pole. The angle [latex]\\theta [\/latex], measured in radians, indicates the direction of [latex]r[\/latex]. We move counterclockwise from the polar axis by an angle of [latex]\\theta [\/latex], and measure a directed line segment the length of [latex]r[\/latex] in the direction of [latex]\\theta [\/latex]. Even though we measure [latex]\\theta [\/latex] first and then [latex]r[\/latex], the polar point is written with the r<\/em>-coordinate first. For example, to plot the point [latex]\\left(2,\\frac{\\pi }{4}\\right)[\/latex], we would move [latex]\\frac{\\pi }{4}[\/latex] units in the counterclockwise direction and then a length of 2 from the pole.\r\n\r\n\"Polar\r\n\r\n
Plot the point [latex]\\left(3,\\frac{\\pi }{2}\\right)[\/latex] on the polar grid.[reveal-answer q=\"457738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"457738\"]The angle [latex]\\frac{\\pi }{2}[\/latex] is found by sweeping in a counterclockwise direction 90\u00b0 from the polar axis. The point is located at a length of 3 units from the pole in the [latex]\\frac{\\pi }{2}[\/latex] direction.\"Polar\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
Plot the point [latex]\\left(2,\\frac{\\pi }{3}\\right)[\/latex] in the polar grid<\/strong>.[reveal-answer q=\"632864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632864\"]\"Polar[\/hidden-answer]<\/section>
Plot the point [latex]\\left(-2,\\frac{\\pi }{6}\\right)[\/latex] on the polar grid.[reveal-answer q=\"158861\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"158861\"]We know that [latex]\\frac{\\pi }{6}[\/latex] is located in the first quadrant. However, [latex]r=-2[\/latex]. We can approach plotting a point with a negative [latex]r[\/latex] in two ways:\r\n
    \r\n \t
  1. Plot the point [latex]\\left(2,\\frac{\\pi }{6}\\right)[\/latex] by moving [latex]\\frac{\\pi }{6}[\/latex] in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;<\/li>\r\n \t
  2. Move [latex]\\frac{\\pi }{6}[\/latex] in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.<\/li>\r\n<\/ol>\r\nCompare this to the graph of the polar coordinate [latex]\\left(2,\\frac{\\pi }{6}\\right)[\/latex].\r\n\r\n\"Two\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    Plot the points [latex]\\left(3,-\\frac{\\pi }{6}\\right)[\/latex] and [latex]\\left(2,\\frac{9\\pi }{4}\\right)[\/latex] on the same polar grid.[reveal-answer q=\"625444\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625444\"]\"Points[\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]174888[\/ohm_question]<\/section>","rendered":"

    Plotting Points Using Polar Coordinates<\/h2>\n

    When we think about plotting points in the plane, we usually think of rectangular coordinates<\/strong> [latex]\\left(x,y\\right)[\/latex] in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates<\/strong>, which are points labeled [latex]\\left(r,\\theta \\right)[\/latex] and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole<\/strong>, or the origin of the coordinate plane.<\/p>\n

    The polar grid<\/strong> is scaled as the unit circle with the positive x-<\/em>axis now viewed as the polar axis<\/strong> and the origin as the pole. The first coordinate [latex]r[\/latex] is the radius or length of the directed line segment from the pole. The angle [latex]\\theta[\/latex], measured in radians, indicates the direction of [latex]r[\/latex]. We move counterclockwise from the polar axis by an angle of [latex]\\theta[\/latex], and measure a directed line segment the length of [latex]r[\/latex] in the direction of [latex]\\theta[\/latex]. Even though we measure [latex]\\theta[\/latex] first and then [latex]r[\/latex], the polar point is written with the r<\/em>-coordinate first. For example, to plot the point [latex]\\left(2,\\frac{\\pi }{4}\\right)[\/latex], we would move [latex]\\frac{\\pi }{4}[\/latex] units in the counterclockwise direction and then a length of 2 from the pole.<\/p>\n

    \"Polar<\/p>\n

    Plot the point [latex]\\left(3,\\frac{\\pi }{2}\\right)[\/latex] on the polar grid.<\/p>\n