Problem 2 - College Trigonometry

Question

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Circle: \(r=2 \cos (\theta)\)

Step-by-Step Solution

Answer

The polar equation \(r = 2 \cos(\theta)\) represents a circle, centered at \((1, 0)\) with a radius of \(1\).

Step 1: Understand the Polar Equation

The given polar equation is \( r = 2 \cos(\theta) \). This equation represents a circle in polar coordinates.

Step 2: Identify the Circle Properties

In polar coordinates, \( r = 2 \cos(\theta) \) describes a circle. This circle is centered on the x-axis, specifically at the point \( r = 1 \) (as the form is \( r = a \cos(\theta) \) where the circle is centered at \((\frac{a}{2}, 0)\) in Cartesian coordinates). The circle has a radius of \(1\).

Step 3: Plot Key Points

Consider four key points for plotting: when \( \theta = 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \). - For \( \theta = 0 \), \( r = 2 \cos(0) = 2 \).- For \( \theta = \frac{\pi}{2} \), \( r = 2 \cos(\frac{\pi}{2}) = 0 \).- For \( \theta = \pi \), \( r = 2 \cos(\pi) = -2 \). In polar coordinates with negative \( r \), the point \((-2, \pi)\) corresponds to \((2, 0)\) directly. It represents the same location since it's on the circle.- For \( \theta = \frac{3\pi}{2} \), \( r = 2 \cos(\frac{3\pi}{2}) = 0 \).

Step 4: Draw the Circle

Starting from the origin, plot the earlier calculated key points and connect them in a smooth circular path. By symmetry and from the given points, the circle is centered at \((1, 0)\) with a radius of \(1\).

Step 5: Label the Graph

Ensure the graph is labeled with the center as \((1, 0)\), and key points such as \((2, 0)\), \((0, 0)\), and \((2, \pi)\) are marked. Note that effectively, these points maintain a distance correctly defining a circle centered on the x-axis.

Key Concepts

Graphing Polar Equations

Polar equations specify a relationship between the radial coordinate, \( r \), and the polar angle, \( \theta \). While in Cartesian coordinates, we express points using \( x \) and \( y \), in polar coordinates, we use \( r \) and \( \theta \). This representation is particularly useful for circular and spiral shapes.
When graphing polar equations like \( r = 2 \cos(\theta) \), we start by comprehending and assessing the behavior of \( \cos(\theta) \) over different values of \( \theta \). This function repeats every \( 2\pi \) and varies between \(-1\) and \(1\).

• Identify key angles where polar graph attributes, such as maximum or symmetry, are showcased: generally \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \).
• Calculate \( r \) for each of these angles to capture important points on the graph.

After plotting these points, connect them to visualize the curve.
In our task, this results in a smooth circle. Always remember to label your graph well, ensuring the center and key points are clear to readers.

Circle in Polar Coordinates

Interpreting circles in polar coordinates can seem complex initially, but it becomes clearer with practice. In polar graphs, circles typically translate into simpler functions compared to Cartesian coordinates.
For the equation \( r = 2 \cos(\theta) \), understanding its representation as a circle is important.

• The circle’s center in Cartesian terms is at \( (\frac{a}{2},0) \), where \( a \) is the coefficient in \( a \cos(\theta) \). Here, \( a = 2 \), so the circle is centered at \( (1,0) \).
• The maximum value for \( r \) when \( \theta = 0 \) is the circle’s farthest point from the origin, \( (2,0) \).

From here, you identify the circle's symmetry along the x-axis. This setup results from its origin-centered alignment in polar form, highlighting polar plots’ ability to simplify certain geometric shapes.

Polar Plot Key Points

In any polar graphing exercise, pinpointing significant angles can simplify plotting and ensure accuracy. These key angles generally include \( 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \), due to their standard positions on the unit circle.
When analyzing \( r = 2 \cos(\theta) \):

• \( \theta = 0 \) gives \( r = 2 \), which is the farthest point on the x-axis.
• \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \) yield \( r = 0 \), indicating the circle intersects the pole (origin).
• \( \theta = \pi \) results in \( r = -2 \), which may seem complex. However, this reflects symmetry relevant in polar plots — understanding negative \( r \) leads us back to the origin (\( (2,0) \)).

This balance simplifies reading a graph from a given polar equation. It's essential to interpret the values flexibly, seeing through the different possibilities a negative \( r \) might imply. With these strategies, polar plots become a viable, thorough graphing method.