Problem 151 - 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. $$ r=\cos \left(\frac{\theta}{5}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Test for Symmetry with respect to the x-axis

To test if a polar equation is symmetric with respect to the x-axis, substitute \(\theta\) with \(-\theta\). For the given equation, this yields:\[r = \cos \left(\frac{-\theta}{5}\right) = \cos \left(-\frac{\theta}{5}\right)\]Since the cosine function is even, \(\cos\left(-x\right) = \cos\left(x\right)\), so:\[\cos \left(-\frac{\theta}{5}\right) = \cos \left(\frac{\theta}{5}\right)\]Thus, \(r = \cos \left(\frac{\theta}{5}\right)\) confirms that the graph is symmetric with respect to the x-axis.

Step 2: Test for Symmetry with respect to the y-axis

To check symmetry with respect to the y-axis, replace \(\theta\) with \(\pi - \theta\). Substitute into the equation:\[r = \cos \left(\frac{\pi - \theta}{5}\right)\]Applying the identity for cosine, \(\cos(\pi - x) = -\cos(x)\), we get:\[r = -\cos \left(\frac{\theta}{5}\right)\]This result, \(r = -\cos\left(\frac{\theta}{5}\right)\), is not equal to the original equation, so the graph is not symmetric with respect to the y-axis.

Step 3: Test for Symmetry with respect to the origin

To test for origin symmetry, replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\). Substitute in the equation:\[-r = \cos \left(\frac{\theta + \pi}{5}\right)\]Using the cosine angle addition formula, \(\cos(x + \pi) = -\cos(x)\), gives:\[-r = -\cos \left(\frac{\theta}{5}\right)\]This simplifies to \(r = \cos \left(\frac{\theta}{5}\right)\), which matches the original equation, confirming symmetry with respect to the origin.

Key Concepts

Polar Coordinates

Polar coordinates are a different way of representing points on a plane, as opposed to the usual Cartesian coordinates, which use the x and y-axis. Here, each point is defined by a distance "r" from the origin and an angle "\(\theta\)" measured from the positive x-axis. This means:

• "r" tells you how far you are from the center, the pole.
• "\(\theta\)" shows the direction or angle you need to move to reach the point.

This can be very useful for certain types of graphs, especially those that have circular or spiral patterns. In polar coordinates, a point can also have many equivalent forms by varying \(\theta\) because angles are periodic. This brings a lot of utility in problems involving symmetry, making it easier to identify symmetries with respect to different axes or the origin.

Trigonometric Symmetry

Understanding symmetry is crucial in identifying how a function behaves when transformed. In polar graphs, this involves checking whether the graph remains unchanged when you rotate it or flip it. Symmetry tests include:

• Symmetry with respect to the x-axis: Substitute \(\theta\) with \(-\theta\). If the equation remains unchanged, the graph is x-axis symmetric.
• Symmetry with respect to the y-axis: Replace \(\theta\) with \(\pi - \theta\). If the equation is the same, the graph is y-axis symmetric.
• Symmetry with respect to the origin: Replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\). If unchanged, it’s origin symmetric.

These transformations allow you to test symmetries based on trigonometric identities like \(\cos(-\theta) = \cos(\theta)\) (even function) and \(\cos(\pi - \theta) = -\cos(\theta)\). Recognizing these symmetries helps in visualizing the graph more easily.

Cosine Function

The cosine function, often denoted as \(\cos(\theta)\), is one of the fundamental trigonometric functions. It's periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians (or 360 degrees). Some key characteristics include:

• Even Function: This means that \(\cos(-x) = \cos(x)\), making symmetry tests simpler.
• Range: The values are strictly between -1 and 1. This makes it ideal for representing amplitudes in wave functions.
• Graph: A smooth wave that starts from 1 at \(\theta=0\), decreases to -1, and returns to 1 at \(2\pi\).

Within polar equations like \(r = \cos(\frac{\theta}{5})\), the cosine function scales and shifts based on the angle division, affecting the spiral and circular symmetry within the graph. Mastering cosine allows for recognizing patterns and symmetries more intuitively in polar plots.