Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

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 given polar equation \(r=5+4 \cos \theta\) is symmetric with respect to \(\theta=\pi / 2\) line and polar axis, but not around the pole.

Step 1: Symmetry Test for \(\theta=\pi / 2\)

To check for symmetry with respect to the line \(\theta=\pi / 2\), replace \(\theta\) with \(-\theta\) in original equation. In this case, equation becomes \(r=5+4 \cos (-\theta)\). As cosine function has property \(\cos(-\theta)=\cos(\theta)\), the equation will return to its original form. Hence, it is symmetric about \(\theta=\pi / 2\) line.

Step 2: Symmetry Test for polar axis

For symmetry about the polar axis, replace \(\theta\) with \(-\theta\). In this case, equation becomes \(r=5+4 \cos (-\theta)\). As cosine function has property \(\cos(-\theta)=\cos(\theta)\), the equation will return to its original form. Hence, it is symmetric about polar axis.

Step 3: Symmetry Test for the pole

To check it, replace \(r\) with \(-r\) and \(\theta\) with \(\theta+ \pi \). The equation does not retain its original state. Hence, it is not symmetric around the pole.

Key Concepts

Understanding Symmetry Tests

Symmetry tests in polar equations help identify the visual symmetry of a graph. These tests make it easier to predict and draw the graph of polar equations. Symmetry with respect to certain lines or points is fundamental for analyzing the shape and behavior of these equations.

• Symmetry with respect to the line \(\theta = \pi/2\): This test detects symmetry about the vertical line in the polar coordinate system. Substitute \(\theta\) with \(-\theta\). If the resulting equation returns to its original form, the graph is symmetric about this line. In our exercise, \(r=5+4 \cos \theta\) becomes \(r=5+4 \cos (-\theta)\), and since \(\cos(-\theta) = \cos(\theta)\), it is symmetric about \(\theta = \pi/2\).


• Symmetry with respect to the polar axis: Here, \(\theta\) is also replaced with \(-\theta\). If the equation simplifies back to the original, the symmetry around the polar axis is confirmed. Again, for our equation, it results in the same, confirming this symmetry.


• Symmetry concerning the pole: Test this by substituting \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\). If the equation does not return to its original form, then there’s no symmetry around the pole. In our example, \(r=5+4 \cos(\theta)\) fails this test, indicating no symmetry at the pole.

These tests are crucial for understanding the structure of polar graphs without plotting every point.

Delving into Polar Coordinates

Polar coordinates provide a unique way of describing the position of a point on a plane, using a combination of a radius and an angle. Unlike Cartesian coordinates, where points are represented by \((x, y)\), polar coordinates are expressed as \((r, \theta)\).

• Radius \(r\): It indicates the distance from the point to the origin, also known as the pole.


• Angle \(\theta\): Represents the rotation from the positive x-axis, measured in radians. \(\theta\) can be positive or negative, with counter-clockwise and clockwise rotations respectively.

Polar coordinates are particularly useful for equations and graphs that involve circular or spiral patterns. They allow functions that are difficult to express in rectangular coordinates to become simpler and more intuitive. Understanding polar coordinates is essential when dealing with equations like the one from our exercise, where \(r = 5 + 4 \cos \theta\) describes a limaçon, a type of curve distinctive in polar graphs.

Role of Trigonometric Functions in Polar Equations

Trigonometric functions such as sine and cosine play a significant role in polar equations, creating connections between angles and radial distances. They help describe the shape and orientation of curves in the polar coordinate system.

• Cosine Functions: In the equation \(r = 5 + 4 \cos \theta\), the cosine function determines how the radius \(r\) changes with different angles \(\theta\). The cyclical nature of cosine, with its known values and symmetry properties, directly impacts the graph's symmetry and shape.


• Sine Functions: While not utilized in the exercise's specific example, sine functions can describe how a curve oscillates or shifts vertically along the polar plane.

By incorporating trigonometric functions, polar equations can depict a wide array of shapes. Understanding these functions' behaviors and properties is crucial for interpreting the resultant graphs accurately. Appreciate how these functions facilitate elegant representations of cycles, symmetries, and periodic patterns.