91Ó°ÊÓ

Determine if the equation has any common polar symmetry properties: symmetry about x-axis, symmetry about y-axis, or symmetry about the origin. To do this, we examine by plugging the following angles into the equation: \(-\theta\), \(\pi - \theta\), and \(\pi + \theta\). Evaluate \(r(-\theta)\): \(r(-\theta) = \cos(-\theta) + \sin(-\theta) = \cos\theta - \sin\theta \neq r(\theta)\) Evaluate \(r(\pi-\theta)\): \(r(\pi-\theta) = \cos(\pi-\theta) + \sin(\pi-\theta) = -\cos\theta + \sin\theta \neq r(\theta)\) Evaluate \(r(\pi+\theta)\): \(r(\pi+\theta) = \cos(\pi+\theta) + \sin(\pi+\theta) = -\cos\theta - \sin\theta \neq r(\theta)\) Thus, the equation has no symmetry properties.

To find the maximum and minimum values of \(r\), we can rewrite the polar equation to \(r = \cos\theta + \sin\theta = \sqrt{2}\sin(\theta + \pi/4)\) by using the angle addition formula and a trigonometric identity. The sine function has a maximum value of 1 and a minimum value of -1, so the maximum \(r\) value is \(\sqrt{2}\) and the minimum is \(-\sqrt{2}\).

Calculate the value of \(r\) at angles \(0\), \(\pi/2\), \(\pi\), and \(3\pi/2\). When \(\theta = 0\): \(r = \cos(0) + \sin(0) = 1 + 0 = 1\) When \(\theta = \pi/2\): \(r = \cos(\pi / 2) + \sin(\pi / 2) = 0 + 1 = 1\) When \(\theta = \pi\): \(r = \cos(\pi) + \sin(\pi) = -1 + 0 = -1\) When \(\theta = 3\pi / 2\): \(r = \cos(3\pi / 2) + \sin(3\pi / 2) = 0 - 1 = -1\)

Using the information we have gathered from steps 1-3, we can now sketch the graph of the polar equation \(r = \cos\theta + \sin\theta\). 1. No symmetry properties. 2. Maximum value of \(r\) is \(\sqrt{2}\) and minimum is \(-\sqrt{2}\). 3. Intersection points at angles \(0\), \(\pi/2\), \(\pi\), and \(3\pi/2\). Start by plotting the intersection points on a polar coordinate system: - At angle \(0\), the graph passes through \(r = 1\). - At angle \(\pi/2\), the graph passes through \(r = 1\). - At angle \(\pi\), the graph passes through \(r = -1\). - At angle \(3\pi/2\), the graph passes through \(r = -1\). Next, draw the graph by connecting these points in a smooth curve, while keeping in mind the maximum and minimum values of \(r\). Finally, review your sketch and make sure it aligns with the information from steps 1-3.