Graphing polar curves: lemniscates

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Graphing polar curves and, more specifically, graphing circles, is a hard thing to master at first. But, with practice, it gets a lot easier, and you'll get much, much faster.

When you have a polar equation, you usually have a trig function in the equation, with an argument inside the trig function that's in terms of theta. The best way to sketch the polar curve is to set that argument inside the trig function equal to pi/2, and then solve that equation for theta. The reason this works is because the values of trig functions along the unit circle at multiples of pi/2 are always 1 or 0, so it makes finding those values really simple.

Once you have those values, you want to mark off a set of cartesian axes with those values of theta, then plot the points (theta,r) on those axes. That'll make it a lot easier to sketch the polar curve on polar axes, because you'll already have an idea of what the graph should look like, and you'll have some specific points already plotted.

Once the cartesian sketch is done, you'll take each of the points you plotted, in order, and move them over to the polar axes one at a time. As you do, you'll see the polar curve start to emerge.

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