91Ó°ÊÓ

To analyze the equation \(r = \cot \theta\), we need to understand the behavior of the cotangent function. The cotangent function, written as \(\cot(\theta)\), is the reciprocal of the tangent function and can be defined as: \[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\] The cotangent function has several characteristics: 1. It is periodic with a period of \(\pi\), meaning \(\cot(x + \pi) = \cot(x)\) for all x 2. It has vertical asymptotes at \(\theta = k\pi\) for any integer k, where the function is undefined 3. It has zeros at \(\theta = k\pi+\frac{\pi}{2}\) for any integer k 4. It approaches infinity as \(\theta\) approaches its vertical asymptotes

Now that we have an understanding of the cotangent function, we need to determine important points and features to help in sketching the polar curve. The curve is defined by the equation \(r = \cot \theta\), which we can rewrite using one of the definitions of cotangent: \[r = \frac{\cos(\theta)}{\sin(\theta)}\] Some important features and points: 1. When \(\theta = 0\), the mathematical function is undefined due to division by zero. This represents a vertical asymptote in the polar coordinate system. 2. At \(\theta = \frac{\pi}{2}\), r = 0, so the curve goes through the pole. 3. The curve approaches infinity as \(\theta\) approaches vertical asymptotes. Using these important points and features, we can now sketch the polar curve.

Using the important points and features found in Step 2, we can sketch the polar curve \(r = \cot \theta\): 1. Draw vertical asymptotes at \(\theta=0\) and other multiples of \(\pi\). Since the cotangent function has a period \(\pi\), these vertical asymptotes repeat at regular intervals. 2. Mark the point (0, \(\frac{\pi}{2}\)) as where the curve intersects the pole. 3. Remember that positive r values represent a distance from the pole in the direction of the angle \(\theta\), while negative r values represent a distance in the opposite direction. 4. As \(\theta\) gets closer to the vertical asymptotes, the curve must approach infinity. So, the graph will have "branches" that get closer to the asymptotes but never intersect them. 5. For \(\theta\) between vertical asymptotes, identify if r is positive or negative and sketch the curve accordingly, following the behavior of the cotangent function. Once all these steps are completed, you'll have a sketch of the polar curve \(r = \cot \theta\). Note that it may take some practice to be able to accurately sketch more complex polar curves.