Problem 150 - Calculus Volume 3

Question

Determine whether the graphs of the polar equation are symmetric with respect to the \(x\)-axis, the \(y\) -axis, or the origin. $$ r^{2}=9 \cos \theta $$

Step-by-Step Solution

Answer

The graph is symmetric with respect to the x-axis and the origin.

Step 1: Testing for symmetry with respect to the x-axis

To check if the graph is symmetric with respect to the x-axis, replace \(\theta\) with \(-\theta\) in the equation. This gives us \(r^2 = 9 \cos(-\theta)\). Since \(\cos(-\theta) = \cos(\theta)\), the equation simplifies back to \(r^2 = 9 \cos(\theta)\). Therefore, the graph is symmetric with respect to the x-axis.

Step 2: Testing for symmetry with respect to the y-axis

Next, replace \(r\) by \(-r\) and \(\theta\) by \(\pi - \theta\). This transforms the equation to \((-r)^2 = 9 \cos(\pi - \theta)\) which simplifies to \(r^2 = 9 (-\cos(\theta))\). This is not equivalent to the original equation. Hence, the graph is not symmetric with respect to the y-axis.

Step 3: Testing for symmetry with respect to the origin

To check for symmetry about the origin, replace \(r\) by \(-r\) and \(\theta\) by \(-\theta\). This gives us \((-r)^2 = 9 \cos(-\theta)\), which simplifies to \(r^2 = 9\cos(\theta)\). This matches the original equation, indicating that the graph is symmetric with respect to the origin.

Key Concepts

Symmetry in Polar Coordinates

Symmetry in polar coordinates involves examining how the graph of a polar equation behaves when it is reflected over certain axes or through the origin. Polar equations are explored using the polar coordinate system, where each point on the plane is determined by an angle, \(\theta\), and a distance from the origin, \(r\).

To determine symmetry, we'll test how transformations affect the equation:

• Symmetry with respect to the x-axis is tested by replacing \(\theta\) with \(-\theta\). If the equation remains unchanged, the graph is symmetric with respect to the x-axis.
• Y-axis symmetry is checked by substituting \(r\) with \(-r\) and \(\theta\) with \(\pi - \theta\). The graph is symmetric if the transformed equation matches the original.
• Origin symmetry is determined by replacing both \(r\) and \(\theta\) with their negatives. Again, if the equation remains the same, the graph is symmetric with respect to the origin.

Testing these symmetries can give insights into the graph's behavior and characteristics.

Graph Symmetry

Graph symmetry refers to the property of a graph to be a mirror image across an axis or a point upon certain transformations. In polar coordinates, symmetry helps simplify the analysis and graphing of equations.

In our example, the equation \(r^2 = 9 \cos \theta\) exhibits graph symmetry as follows:

• The graph is symmetric with respect to the x-axis, since substitution \(\theta\) with \(-\theta\) results in the identical equation.
• No symmetry is found with respect to the y-axis as substituting \(\pi - \theta\) and changing \(r\) to \(-r\) does not retain the equation's form.
• The graph is symmetric about the origin as substituting both \(r\) and \(\theta\) with their negatives returns the original equation.

Recognizing graph symmetry assists in predicting graph patterns and reducing graphing efforts by focusing on just one symmetric part.

Polar Coordinate System

The polar coordinate system represents a two-dimensional plane in terms of distance and angle from a fixed central point known as the origin. Unlike the Cartesian coordinate system that uses \(x\) and \(y\) positions, the polar system specifies points using the radius \(r\), the distance from the origin, and \(\theta\), the angle from the positive x-axis.

Benefits of Polar Coordinates:

• They are particularly effective for describing circular and radial patterns, where equations and graphs that exhibit symmetry can be more easily manipulated.
• Polar forms allow for the elegant simplification of equations involved in circular geometry, which might otherwise be complex in rectangular coordinates.

Exploring Polar Equations Further:
When dealing with polar equations, identifying symmetries helps to understand the graph's form and behavior. This insight is beneficial in applications like navigation and physics, which frequently involve such geometrical considerations.