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Figure \(\PageIndex{3}\) shows a point \(P\) in the plane with rectangular coordinates \((x,y)\) and polar coordinates \(P(r,\theta)\). Using trigonometry, we can make the identities given in the following Key Idea.
Learn how to use polar coordinates to describe points in the plane and graph functions. This web page covers the basics of polar coordinates, conversion between polar and rectangular coordinates, and graphs of polar functions.
- Testing a Polar Equation for Symmetry. Test the equation r=2sinθ r=2sinθ for symmetry. Solution. Test for each of the three types of symmetry. 1) Replacing (r,θ) (r,θ) with (−r,−θ) (−r,−θ) yields the same result.
- Finding Zeros and Maximum Values for a Polar Equation. Using the equation in Example 1, find the zeros and maximum | r | | r | and, if necessary, the polar axis intercepts of r=2sinθ.
- Sketching the Graph of a Polar Equation for a Circle. Sketch the graph of r=4cosθ. r=4cosθ. Solution. First, testing the equation for symmetry, we find that the graph is symmetric about the polar axis.
- Sketching the Graph of a Cardioid. Sketch the graph of r=2+2cosθ. r=2+2cosθ. Solution. First, testing the equation for symmetry, we find that the graph of this equation will be symmetric about the polar axis.
To find the coordinates of a point in the polar coordinate system, consider Figure \(\PageIndex{1}\). The point \(P\) has Cartesian coordinates \((x,y)\). The line segment connecting the origin to the point \(P\) measures the distance from the origin to \(P\) and has length \(r\).
Sep 17, 2022 · These equations demonstrate how to find the Cartesian coordinates when we are given the polar coordinates of a point. They can also be used to find the polar coordinates when we know \(\left( x, y \right)\).
Mar 22, 2024 · Convert the coordinate plane to a polar grid with just a pair of clicks, then you’re free to explore the beauty of circles, spirals, roses, limacons and more in this polar graphing playground. Get started with the video on the right, then dive deeper with the resources and challenges below.