Problem 91 - Precalculus

Question

Given \(r=1+3 \cos \theta,\) find the \(\theta\) -intervals for the inner loop.

Step-by-Step Solution

Answer

The \( \theta \)-interval for the inner loop is \( (1.910, 4.373) \).

Step 1: Understanding the Problem

The given equation is in polar coordinates, where \( r = 1 + 3\cos\theta \). We need to find the \( \theta \) intervals where the graph of this polar equation has an inner loop. The inner loop occurs when \( r < 0 \).

Step 2: Set Up the Inequality

To find when the inner loop occurs, we solve the inequality \( 1 + 3\cos\theta < 0 \).

Step 3: Solve the Inequality

Rearranging the inequality gives \( 3\cos\theta < -1 \). Dividing through by 3 simplifies this to \( \cos\theta < -\frac{1}{3} \).

Step 4: Find the Angles

The inequality \( \cos\theta < -\frac{1}{3} \) corresponds to angles \( \theta \) where the cosine function is less than \(-0.333\). Using a calculator or unit circle and considering the range of cosine, angles are in the second and third quadrants. Find the reference angle from \( \cos^{-1}\left(\frac{1}{3}\right) \approx 1.231\) radians. Thus, \( \theta \) ranges from \( 2\pi - 1.231 \approx 5.052\) to \( \pi + 1.231 \approx 4.373\).

Step 5: Express the Interval for \( \theta \)

Thus, the \( \theta \) intervals where the inner loop exists are \( \theta \in (1.910, 4.373) \) or approximately in degrees, \( \theta \in (109.5^\circ, 250.5^\circ) \).

Key Concepts

Inner Loop

In polar graphing, an inner loop is a fascinating feature that occurs in some equations. It is part of the curve where the radius, denoted by \( r \), becomes negative. This results in a loop that appears inside the main shape of the graph.
For the equation \( r = 1 + 3\cos\theta \), we identify the presence of an inner loop by checking where \( r < 0 \). This indicates that part of the graph "loops" back on itself. Understanding this concept is vital when analyzing polar equations to determine where these loops occur.
Inner loops are especially common in limaçons, a type of curve that may have an inner loop, a dimple, or appear as a convex shape, depending on specific parameters in the equation.

Inequality

An inequality is a mathematical statement that relates expressions that are not equal. In this context, we use inequalities to find the conditions under which certain features like inner loops occur in polar graphs.
Specifically, for \( r = 1 + 3\cos\theta \), the inequality \( 1 + 3\cos\theta < 0 \) is solved to find the \( \theta \)-intervals that create the inner loop.

• First, rearrange the inequality to isolate the cosine component: \( 3\cos\theta < -1 \).
• Divide by 3 to simplify: \( \cos\theta < -\frac{1}{3} \).

Solving this inequality helps determine the exact angles where the loop crosses the pole (origin of the graph) and becomes negative.

Cosine Function

The cosine function is a fundamental part of trigonometry and plays a crucial role in polar coordinates. It describes oscillations typically starting at its maximum values and decreasing, shaping the graph of polar equations significantly.
For \( r = 1 + 3\cos\theta \), the cosine function affects how \( r \) varies as \( \theta \) changes. The key is understanding how cosine values affect the radius:

• Maximum value: when \( \cos\theta = 1 \).
• Minimum value: when \( \cos\theta = -1 \).

The inequality \( \cos\theta < -\frac{1}{3} \) shows us that the cosine is negative and quite small, which in turn generates an inner loop for the specified \( \theta \) intervals.

Polar Equation Graph

Polar equation graphs are unique representations that use radial distances from a central point based on angles. These graphs provide a distinct method of visualizing relationships, different from traditional Cartesian coordinates.
For \( r = 1 + 3\cos\theta \), the graph indicates how the radius \( r \) changes as the angle \( \theta \) changes, revealing dynamic variations like loops and dimples. Key components include:

• The shape: affected by changes in \( r \) relative to \( \theta \).
• Intervals: specific angles which detail unique features like inner loops.

Learning to interpret these graphs allows for better comprehension of complex visual data represented in polar coordinates.