Use the conversion formulas to convert equations between rectangular and polar coordinates. Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.

Sketch polar curves from given equations; Convert equations between rectangular and polar coordinates; Identify symmetry in polar curves and equations

By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of r) to determine the graph of a polar equation. In the first test, we consider symmetry with respect to the line θ = π 2 (y -axis).

Understanding the symmetry of polar curves. Polar curves may exhibit symmetry over the polar axis, the line $\boldsymbol{\theta = \dfrac{\pi}{2}}$, or the pole. Once we establish the rules of symmetry for a polar equation, we can evaluate fewer polar coordinates and simply reflect half the curve over whichever it exhibits symmetry on.

A graph has symmetry with respect to the x-axis if, whenever (x, y) is on the graph, so is the point (x, -y). The big question is how do we test for symmetry of an equation in polar coordinates?

1. Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane. 2. Which of the three types of symmetries for polar graphs correspond to the symmetries with respect to the x-axis, y-axis, and origin? 3. What are the steps to follow when graphing polar equations? 4.

By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of [latex]r [/latex]) to determine the graph of a polar equation.

2023年5月19日 · A polar graph is symmetric with respect to $r=\frac{\pi}{2}$ [the $y$-axis] if replacing $(r,\theta)$ with either $(r,\pi-\theta)$ or $(-r,-\theta)$ gives an equivalent equation. $r=\cos(2(\pi-\theta))=\cos(2\theta)$ so the polar graph is symmetric with respect to the $y$-axis.

Polar symmetry refers to the property of a graph or equation in polar coordinates that remains unchanged when rotated by 180 degrees about the origin. This type of symmetry is significant in understanding the geometric features of polar equations …

Some graphs have symmetry and which makes allows you to use fewer points to make the graph. There are three types of symmetry: line \(θ = \frac{π}{2}\) (y-axis), polar axis (x-axis), and pole (origin). Each of these has a test to determine if a polar equation has that type of symmetry.