Problem 13 - Precalculus

Question

Test for symmetry with respect to the line \(\theta=\pi 2,\) the polar axis, and the pole. $$r=5+4 \cos \theta$$

Step-by-Step Solution

Answer

The equation \( r=5+4 \cos \theta \) is symmetrical about the polar axis but it is not symmetrical about the pole nor the line \( \theta=\pi/2 \).

Step 1: Testing for symmetry about the polar axis

Substitute \( \theta \) by \( -\theta \) in the function \( r=5+4 \cos \theta \) to check if it remains the same. The new function is \( r=5+4 \cos (-\theta) \). Using the fact that \( \cos(-\theta) = cos(\theta) \), we get back \( r=5+4 \cos \theta \). Therefore, the equation exhibits symmetry about the polar axis.

Step 2: Testing for symmetry about the pole

Substitute \( \theta \) by \( \theta+\pi \) in the function \( r=5+4 \cos \theta \). The new function is \( r=5+4 \cos (\theta+\pi) \). Using the fact that \( \cos(\theta+\pi) = -cos(\theta) \), we do not get back the same equation, \( r=5+4 \cos \theta \). Therefore, the equation doesn't exhibit symmetry about the pole.

Step 3: Testing for symmetry about the line \( \theta=\pi/2 \)

Substitute \( r=-r \) and \( \theta=\pi - \theta \) in \( r=5+4 \cos \theta \). The new function is \( -r=5+4 \cos (\pi-\theta) \). Using the fact that \( \cos(\pi-\theta) = -cos(\theta) \), we do not get back the same equation, \( r=5+4 \cos \theta \). Therefore, the equation doesn't exhibit symmetry about the line \( \theta=\pi/2 \).

Key Concepts

Polar Coordinates

When we talk about polar coordinates, we refer to a system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is known as the pole, which is similar to the origin in the Cartesian coordinate system, and the reference direction is usually the positive x-axis, referred to as the polar axis.

In polar coordinates, a point is represented as \( (r, \theta) \), where \( r \) is the radial distance from the pole and \( \theta \) is the angle measured in radians from the polar axis. This system is particularly useful for graphs involving circles or other symmetrical shapes, and it is extensively used in fields such as physics, engineering, and navigation.

Cosine Symmetry

Cosine symmetry is a property often discussed within the context of trigonometric functions and polar coordinates. A polar equation exhibits cosine symmetry if replacing \( \theta \) with \( -\theta \) yields the same function. This is due to the even nature of the cosine function, which holds that \( \cos(-\theta) = \cos(\theta) \).

Exploring Cosine Symmetry

In the context of the given problem \( r=5+4 \cos \theta \), one can observe cosine symmetry because substituting \( -\theta \) for \( \theta \) results in no change to the equation, showing that this polar equation is symmetric about the polar axis. It means the shape remains unchanged when reflected over this axis.

Precalculus

Precalculus serves as the foundational course that bridges algebra and calculus, covering a variety of topics such as functions, complex numbers, matrices, sequence and series, and trigonometry. Understanding polar coordinates and symmetry is a significant precalculus concept, as it lays the groundwork for more advanced studies in calculus and mathematics as a whole.

Within precalculus, students learn about the behavior of different functions and how to identify symmetries which can simplify graphing and understanding the properties of equations. Being proficient in these areas allows students to tackle higher-level math with a solid ground in how to navigate different coordinate systems and identify patterns in equations.

Polar Axis Symmetry

Polar axis symmetry occurs when a polar graph remains unchanged upon reflection across the polar axis. Graphically, this means that for every point with coordinates \( (r, \theta) \) on the graph, there exists a corresponding point with coordinates \( (r, -\theta) \) that is also on the graph.

In terms of polar equations, one can determine polar axis symmetry by substituting \( -\theta \) for \( \theta \) and checking if the original equation is obtained. If so, the polar equation exhibits symmetry about the polar axis, which was demonstrated in the solution of the exercise \( r=5+4 \cos \theta \) by showing the equation remains unchanged under this substitution.

Polar Equations

Polar equations are mathematical expressions that define the relationship between the radius \( r \) and the angle \( \theta \) in polar coordinates. They are used to describe curves and shapes in the polar coordinate system, often revealing symmetries or patterns not immediately apparent in their Cartesian form. Common forms of polar equations include \( r = f(\theta) \) where \( f \) can be any function involving trigonometric, exponential, or algebraic expressions.

Analyzing Polar Equations for Symmetry

To analyze a polar equation for different kinds of symmetry, such as symmetry with respect to the pole or a specific line in the polar plane, one can manipulate the equation algebraically using established symmetry tests. This involves replacing variables according to the type of symmetry under investigation and checking whether the original equation is retrieved. The absence or presence of symmetry can provide insights into the nature of the graph produced by the polar equation.