Problem 16 - College Trigonometry

Question

Graph the following equations. $$ r=\frac{6}{3-\cos \left(\theta+\frac{\pi}{4}\right)} $$

Step-by-Step Solution

Answer

Graph the ellipse centered at \( \theta = -\pi/4 \) with \( e = 1/3 \) using symmetry and key polar coordinates.

Step 1: Identify the Polar Equation Type

The given equation is a form of the polar equation representing a conic section. It can be written in the form of \( r = \frac{ed}{1 - e\cos(\theta-\theta_0)} \) where \( e \) is the eccentricity. The general form indicates it is a conic section.

Step 2: Determine Conic Section Type

The equation has the form \( r = \frac{ed}{1 - e\cos(\theta + \alpha)} \), which can be rewritten as \( r = \frac{6}{-\cos(\theta + \pi/4) + 3} \). Divide numerator and denominator by 3, yielding \( r = \frac{2}{1 - \frac{1}{3}\cos(\theta + \pi/4)} \). Thus, \( e = \frac{1}{3} \), indicating it is an ellipse, since \( e < 1 \).

Step 3: Identify Key Features of the Ellipse

For an ellipse, the center is defined around the angle \( \theta_0 = -\pi/4 \). The directrix is perpendicular to the major axis, and the parameter \( d \) (numerator) is 2, implying the semi-major axis' length is related to \( d \) and \( e \). Check for symmetry around \( \theta = \theta_0 \).

Step 4: Sketching the Graph

Plot the graph considering that this is an ellipse centered at \( \theta_0 = -\pi/4 \). Since \( e = 1/3 \), the ellipse is less eccentric hence more circular. Begin plotting points for various \( \theta \) values and calculate \( r \) to identify the shape accurately. Consider key \( \theta \) angles such as \( 0, \pi/2, \pi, 3\pi/2 \) along with others to ensure an accurate plot.

Step 5: Analyze Symmetry and Adjustments

Use symmetry properties around \( \theta_0 = -\pi/4 \) to verify plotted points maintain elliptical form. Adjust for any asymmetries and ensure graph is centered around the expected directrix perpendicular to the major axis, confirming the overall dimensions and orientation.

Key Concepts

Conic Sections

In mathematics, conic sections are the curves obtained by intersecting a plane with a double-napped cone. Conic sections include ellipses, parabolas, and hyperbolas, each having distinct properties.
Conic sections can be described using polar coordinates with equations that take the form of \[ r = \frac{ed}{1 - e\cos(\theta-\theta_0)} \]

• If the eccentricity \(e = 0\), the conic section is a circle.
• If \(0 < e < 1\), it forms an ellipse.
• If \(e = 1\), it represents a parabola.
• If \(e > 1\), it forms a hyperbola.

This makes conic sections versatile in describing various physical and geometric phenomena. They are fundamental in fields like astronomy, physics, and engineering.
Each conic section has specific focusing properties. For example, ellipses have two focal points, while parabolas and hyperbolas have distinct focal characteristics, impacting their mathematical and geometric behavior.

Eccentricity in Polar Coordinates

Eccentricity \(e\) is a measure of how much a conic section deviates from being a circle. In polar coordinates, eccentricity helps determine the type of conic through the equation form: \[ r = \frac{ed}{1 - e\cos(\theta-\theta_0)} \]In this form:

• \(e\) represents the eccentricity.
• \(d\) is the semi-directrix, influencing the size.

Eccentricity can significantly affect the shape of the conic section:
- **If \(e = 0\):** The shape is a perfect circle, as all points are equidistant from a center point.
- **If \(0 < e < 1\):** The shape is an ellipse with an elongated form, as seen in our exercise where \(e = \frac{1}{3}\). The closer \(e\) is to 0, the more circular the ellipse.
- **If \(e = 1\):** The shape transforms into a parabola, extending infinitely in one direction.
- **If \(e > 1\):** The shape becomes a hyperbola, splitting into two separate curves.
Using eccentricity, we can conclude crucial properties such as symmetry, size, and orientation, providing insights into the graphing process.

Sketching Ellipses

To sketch an ellipse using polar coordinates, it's vital to understand both the mathematical form and the geometric properties. Starting from the polar form,\[ r = \frac{ed}{1 - e\cos(\theta-\theta_0)} \]we identify parameters like eccentricity \(e\), and the reference angle \(\theta_0\).
Begin by setting up a coordinate plane, then identify \(\theta_0\), the angle around which the ellipse is centered. The curve behavior rotates around this angle, contributing to symmetry. For an ellipse, the cosine component shifts the focus location and dictates where the curve bends.
Consider these steps to accurately sketch:

• Identify key angles such as \(0\), \(\pi/2\), \(\pi\), and \(3\pi/2\) to understand symmetry.
• Calculate radius \(r\) for several \(\theta\) values, plotting these points.
• Use symmetry properties to estimate additional points and ensure the ellipse follows a smooth path.
• Adjust and redefine points for neatness and accuracy, ensuring alignment with the semi-major and semi-minor axes.

In the example exercise, \(e = \frac{1}{3}\) ensures the ellipse is fairly circular, emphasizing uniformity around \(\theta_0 = -\pi/4\). Recognizing these components guides effective and precise sketching, creating accurate representations in polar forms.