Problem 38 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=4 \cos \theta+4 \sin \theta$$

Step-by-Step Solution

Answer

The graph of the equation \(r=4\cos\theta+4\sin\theta\) is symmetric about the origin, but not about the x-axis or y-axis.

Step 1: Testing for symmetry around the x-axis

For symmetry around the x-axis, replace \(\theta\) by \(-\theta\). If the equation remains the same or equivalent, then it's symmetric about the x-axis. That means substituting \(\theta\) with \(-\theta\):\[r=4\cos(-\theta)+4\sin(-\theta)=4\cos\theta-4\sin\theta\]The resulting equation is not the same as the original one, therefore, the graph isn't symmetric about the x-axis.

Step 2: Testing for symmetry around the y-axis

For determining symmetry around the y-axis, replace \(\theta\) with \(\pi-\theta\) in the equation. If the equation remains the same, then it's symmetric around the y-axis. So, plugging \(\pi-\theta\) into the given equation returns:\[r = 4\cos(\pi-\theta)+4\sin(\pi-\theta)= -4\cos\theta+4\sin\theta\]This equation is different from the original, hence the graph isn't symmetric about the y-axis.

Step 3: Testing for symmetry about the origin

Finally, for the origin symmetry test, replacing \(r\) and \(\theta\) by \(-r\) and \(\theta+\pi\) accordingly. If this leaves the equation unchanged, it's symmetric about the origin. After applying the replacements, \[-r = 4\cos(\theta+\pi)+4\sin(\theta+\pi)\]which simplifies to \[-r = -4\cos\theta-4\sin\theta\].Multiplying by \(-1\), we obtain the original equation. Therefore, the graph is symmetric about the origin.

Step 4: Graphing the equation

To draw the graph in polar coordinates, plot a series of points for various \(\theta\) values and then connect these points smoothly. Various software packages can be used for this purpose, or it can be sketched by hand.