Sketching Polar Curves

$Sketching Polar Curves Example 1$

Unless after lots of practice you become comfortable sketching polar curves directly onto a polar coordinate system, it’s much easier to sketch the graph first on an x/y-coordinate plane, or cartesian coordinate system, and then transfer the graph from the cartesian system to the polar system.

In order to graph the polar equation on a cartesian system, the first thing you should do is take the value inside the trigonometric identity in your equation, and set that equal to $\frac{pi}{2}$ . For example, if your polar equation is $r=\sin{2\theta}$ , then set $2\theta$ equal to $\frac{pi}{2}$ . Then solve for $\theta$ . Whatever you get for $\theta$ , mark off the $x$ -axis of your cartesian coordinate system in increments of $\theta$ .

The purpose of doing this is that it will make it easier to evaluate your polar equation of each of the points you just marked off on the $x$ -axis. Your next step is to evaluate your polar equation at each of the points at which you just marked off on your $x$ -axis, starting with $x=0$ . Plug each of these points into your polar equation, and plot the resulting value of the polar equation on your cartesian coordinate system.

Once you have four to six points plotted on your cartesian coordinate system, you can start to think about plotting them onto a polar coordinate system. Keep in mind that the $x$ value from your cartesian coordinate system will translate as $\theta$ on your polar coordinate system, which is the angle of rotation. The $y$ value from your cartesian coordinate system will translate as $r$ on your polar coordinate system, which is the distance from the origin, at the angle $\theta$ .

Once you’ve translated each of the $(x,y)$ coordinates into $(r,\theta)$ coordinates, connect the dots to sketch the graph of your polar equation.

Sketching Polar Curves

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