Problem 9 - College Trigonometry

Question

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs.In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Cardioid: \(r=3-3 \cos (\theta)\)

Step-by-Step Solution

Answer

Plot the graph as a cardioid that starts and ends at the origin, reaching its maximum distance at \(r=6\) for \(\theta = \pi\).

Step 1: Understanding the Polar Equation

The given polar equation is \(r = 3 - 3 \cos(\theta)\). This is the equation of a cardioid, which is a type of limaçon. In this form, the parameter '3' outside the cosine function affects the distance, and '3' inside the cosine affects the shape and orientation of the cardioid.

Step 2: Calculate Key Points

We begin by calculating values of \(r\) for key angles \(\theta\): 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and 2\(\pi\). For \(\theta = 0\), \(r = 3 - 3\cos(0) = 0\). For \(\theta = \frac{\pi}{2}\), \(r = 3 - 3\cos\left(\frac{\pi}{2}\right) = 3\). For \(\theta = \pi\), \(r = 3 - 3\cos(\pi) = 6\). For \(\theta = \frac{3\pi}{2}\), \(r = 3 - 3\cos\left(\frac{3\pi}{2}\right) = 3\). For \(\theta = 2\pi\), \(r = 3 - 3\cos(2\pi) = 0\).

Step 3: Plot the Calculated Points

On the polar grid, plot the points calculated: For \(\theta = 0\), place a point at the origin with \(r = 0\). For \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), mark points at radius 3. For \(\theta = \pi\), place a point at \(r = 6\), on the opposite side of the pole from \(\theta = 0\).

Step 4: Connect the Points with a Smooth Curve

Starting from the origin at \(\theta = 0\), draw a smooth curve through each of the calculated points. The curve should extend outwards to the maximum radius of \(6\) at \(\theta = \pi\) then loop back to meet the origin again at \(\theta = 2\pi\), forming a heart-shaped cardioid.

Step 5: Label the Graph

Label your graph, indicating each key point you calculated as well as the axis (0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), 2\(\pi\)). Ensure that your labels clearly show the orientation and key features of the cardioid.

Key Concepts

Understanding Cardioid in Polar Coordinates

A cardioid in polar coordinates is a fascinating shape resembling a heart or an apple. Its equation typically takes the form \( r = a + a \cos(\theta) \) or \( r = a + a \sin(\theta) \). In your exercise, the equation is \( r = 3 - 3 \cos(\theta) \), which is a specific type of cardioid derived from the general shape of a limaçon.

The characteristics of this cardioid include:

• The '3' outside the cosine term in the equation impacts the size of the cardioid. It defines the maximum extent of the curve.
• The negative sign before the cosine function results in symmetry about the x-axis, often creating a distinct loop or indentation in the shape.
• Cardioids can exhibit different shapes depending on whether cosine or sine functions are used, altering their orientation.

Understanding cardioids help illustrate symmetry and loops in polar graphs, offering unique insights into trigonometric functions and their graphical representations.

Exploring the Polar Grid

The polar grid is an essential tool when plotting polar equations like the cardioid. Unlike the Cartesian coordinate system, where points are determined by \(x\) and \(y\) values, the polar grid uses radius \(r\) and angle \(\theta\) to determine a point's position.

Here are some key points about the polar grid:

• The origin in polar coordinates, also called the pole, is where \(r = 0\).
• Angles are measured in radians from the polar axis, akin to the positive x-axis in Cartesian coordinates.
• The radius \(r\) indicates how far the point is from the pole.
• Points for different values of \(\theta\) reveal the path traced by a polar equation, aiding in visualizing complex curves like the cardioid.

Mastering the polar grid can simplify understanding the behavior of polar equations, enhancing intuitive grasp of these mathematical concepts.

Limaçon Shapes and their Diversity

Limaçons are part of a broader family of curves in polar coordinates, closely related to the cardioid. The general form of a limaçon is \( r = a + b \cos(\theta) \) or \( r = a + b \sin(\theta) \), where the cardioid is a special case when \( a = b \).

Here's what makes limaçons intriguing:

• Depending on the ratio between \( a \) and \( b \), limaçons can vary in shape considerably. They can have inner loops, look like circles, or even appear as distorted circles.
• If \( a = b \), the result is a cardioid, demonstrating the transitional phase between a standard limaçon and this distinct curve.
• When \( a < b \), limaçons develop inner loops, making their visual interpretation even more complex and fascinating.

Limaçons and their unique shapes offer diverse examples of how polar equations can generate intricate curves, enriching the study of mathematics with astonishing visual representations.