Problem 96 - Precalculus

Question

In calculus, when we need to find the area enclosed by two polar curves, the first step consists of finding the points where the curves coincide. Find the points of intersection of the given curves. $$r=\cos \theta \quad \text { and } \quad r=2+3 \cos \theta$$

Step-by-Step Solution

Answer

The curves intersect at \((r, \theta) = (-1, \pi)\).

Step 1: Set Equations Equal

We start by setting the two given polar equations equal to each other to find their intersection points. Thus, we have:\[ \cos \theta = 2 + 3 \cos \theta \]

Step 2: Rearrange Equation

Rearrange the equation to bring similar terms on one side:\[ \cos \theta - 3 \cos \theta = 2 \]\[ -2 \cos \theta = 2 \]

Step 3: Solve for cos θ

Divide both sides by -2 to solve for \( \cos \theta \):\[ \cos \theta = -1 \]

Step 4: Find θ Values

The value of \( \theta \) for which \( \cos \theta = -1 \) is \( \theta = \pi \) (or an equivalent angle like \(-\pi\), etc., within the polar coordinate context).

Step 5: Find Corresponding r Values

Substitute \( \theta = \pi \) back into one of the original equations to find \( r \). Using \( r = \cos \theta \), we have:\[ r = \cos(\pi) = -1 \].

Key Concepts

Points of Intersection

Finding points of intersection for two curves in the polar coordinate system is crucial when analyzing their relationship. In polar coordinates, each point is defined by a radius \( r \) and an angle \( \theta \). To find where two curves intersect, we need to determine when both curves have the same \( r \) and \( \theta \).

In the given problem, the curves are \( r = \cos \theta \) and \( r = 2 + 3 \cos \theta \).
To find their intersections, we set the equations equal to each other: \( \cos \theta = 2 + 3 \cos \theta \).
Rearranging the equation gives \( -2 \cos \theta = 2 \), so \( \cos \theta = -1 \).

Solving \( \cos \theta = -1 \) gives \( \theta = \pi \) (and angles equivalent to \( \pi \) like \(-\pi\)).
Substitute back into the original equation to confirm the value of \( r \):

• Using \( r = \cos(\pi) \), we find \( r = -1 \).
• This means the curves intersect at \( (r, \theta) = (-1, \pi) \).

These points are crucial to understanding where the curves might enclose an area.

Area Enclosed by Curves

Calculating the area enclosed by two polar curves usually involves finding their points of intersection first. Once we know where curves intersect, we can integrate to find the area between them. Polar coordinates simplify the integration process when dealing with circles and other round shapes.

To find the area enclosed between two curves, we often use this formula:

• \[ A = \frac{1}{2} \int_{\theta_1}^{\theta_2} (r_2^2 - r_1^2)\, d\theta \]

Here, \( r_1^2 \) and \( r_2^2 \) are expressions for the radii of the two curves at each angle \( \theta \).
\( \theta_1 \) and \( \theta_2 \) are the angles that define the section of interest.

In our example, knowing the intersection points help identify \( \theta_1 \) and \( \theta_2 \), and allow us to set the correct limits for integration.
This approach accurately calculates not only simple symmetric areas but also more complex regions by appropriately choosing and solving for these intersection points.

Trigonometric Equations

Trigonometric equations often arise when working with polar coordinates, as it involves circular functions. These equations involve angles but solve in a similar way to algebraic equations.

In the given problem, solving for where \( \cos \theta = -1 \) means finding angles \( \theta \) such that the cosine function equals a specific value.
This particular equation is simple and well-known:

• \( \cos \theta = -1 \) results in \( \theta = \pi + 2n\pi \), where \( n \) is any integer to capture all possible instances.

This cyclical nature of trigonometric functions is the key because it allows the examination of angles across full circles, recognizing periodicity.

More complex trigonometric equations might involve identities or transformations to simplify and solve for \( \theta \, or \) other intervals.
Understanding how to manipulate trigonometric equations lets us determine key angles that describe behavior in polar coordinates.