![]() (This one is called a cardioid because it is heart-shaped. We see that our equation in polar coordinates, r = 3 cos 2 θ, is much simpler than the rectangular equivalent. So `r = 3 cos\ 2θ` in polar coordinates is equivalent Taking the positive square root of `r^2=x^2+y^2` gives us: Now since `cos\ theta=x/r`, `sin\ theta=y/r` and `r^2=x^2+y^2`, we have To convert `r = 3\ cos\ 2θ` into rectangular coordinates, we use the fact that Why? We convert the function given in this question to rectangular coordinates to see how much simpler it is when written in polar coordinates. Next, here's the answer for the conversion to rectangular coordinates. Conversion from Polar to Rectangular Coordinates Notice the curve is fully drawn once θ takes all values between 0 and 2 π. Clearly, we would need to calculate more than this number of points to get a good sketch. I have only plotted the first 7 points above to keep the graph simple. Recall: A negative " r" means we need to be on the opposite side of the origin. I have drawn arrows to indicate the basic direction we have to head in to get to the next point. We start at Point 1, (3, 0°), and move around the graph by increasing the angle and changing the distance from the origin (determined by substituting the angle into r = 3 cos 2 θ. Placing those first 7 points on a polar coordinate grid gives us the following: The grapher appends a suitable interval to function expressions and graphs them on the specified domain.You can change the endpoints, but they must be finite for graphing functions in the polar coordinate system.The polar function grapher automatically changes infinite values to finite ones. I've put degrees and the radian equivalents. You'll need to set up a table of values, as follows. What if you can't use a computer to draw the graph? See (Figure), (Figure), and (Figure).Using a Table of Values to Sketch this Curve
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