Problem 42 - Precalculus with Calculus Previews

Question

Find the points of intersection of the graphs of the given pair of polar equations. $$ r=3-3 \cos \theta, r=3 \cos \theta $$

Step-by-Step Solution

Answer

The points of intersection are \( \left( \frac{3}{2}, \frac{\pi}{3} \right) \) and \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \).

Step 1: Understanding the given polar equations

We begin with two polar equations: \( r = 3 - 3 \cos \theta \) and \( r = 3 \cos \theta \). Our task is to find the values of \(r\) and \(\theta\) for which both equations are satisfied, indicating points of intersection.

Step 2: Equating the two polar equations

Set the two equations equal to each other to find their intersection points: \[ 3 - 3 \cos \theta = 3 \cos \theta \] The goal here is to solve for \( \theta \).

Step 3: Solving for \( \cos \theta \)

Rearrange the equation: \[ 3 = 6 \cos \theta \] Solve for \( \cos \theta \) by dividing both sides by 6: \[ \cos \theta = \frac{1}{2} \].

Step 4: Finding \(\theta\) values

Identify the angles \(\theta\) where \(\cos \theta = \frac{1}{2}\). These are \(\theta = \frac{\pi}{3} \) and \(\theta = \frac{5\pi}{3} \) since the cosine function is positive in the first and fourth quadrants.

Step 5: Calculate corresponding \(r\) values

Substitute the values of \(\theta\) back into either polar equation, here we'll use \( r = 3 \cos \theta \): - For \( \theta = \frac{\pi}{3} \), \( r = 3 \times \frac{1}{2} = \frac{3}{2} \).- For \( \theta = \frac{5\pi}{3} \), \( r = 3 \times \frac{1}{2} = \frac{3}{2} \).

Step 6: Listing the points of intersection

The points of intersection in polar coordinates are: - \( \left( \frac{3}{2}, \frac{\pi}{3} \right) \)- \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \).

Key Concepts

Points of Intersection

When we talk about points of intersection in polar equations, we're looking for where two polar curves cross each other. Think of it like two roads that meet at a junction. In our example, when we have the equations \( r = 3 - 3 \cos \theta \) and \( r = 3 \cos \theta \), we want to know where they share the same \( (r, \theta) \), which means they meet or intersect.

To find this shared point, we equate the equations to see where they overlap. Finding such points not only helps in plotting the curves accurately on a polar graph but also in understanding how these curves relate spatially. Identifying all intersection points ensures no information is lost and enhances our graph comprehension.

Solving Polar Equations

Solving polar equations may sound challenging, but with some algebra and trigonometry, it's pretty straightforward. Our approach for finding intersections involves setting two given polar equations equal and isolating for \( \theta \).

First, rearrange the combined equation to solve for trigonometric functions, like \( \cos \theta \), which can be less intuitive than the usual x-y setup but follow similar trigonometric identities. Solve for these trigonometric identities to find various possible angles \( \theta \) within a typical range \( [0, 2\pi] \).

Once you have values for \( \theta \), substitute back into one of the original polar equations to find the corresponding \( r \). In our case, both curves had \( \theta \) values where \( \cos \theta = \frac{1}{2} \), giving us the points of intersection.

Polar Coordinates

Polar coordinates are a unique way to map points, different from regular Cartesian coordinates. Instead of using x and y, we use \( r \) (radial distance) and \( \theta \) (angle). This is particularly useful for curves and functions centered around a point or originating radially.

To understand polar graphs, remember:

• \( r \) is the distance from the origin. If \( r > 0 \), it's outward from the pole; if \( r < 0 \), it's inward.
• \( \theta \) is the angle from the positive x-axis, usually measured counterclockwise.

If you visualize a circle graph, radiating outward lines at different angles forms different \( \theta \). The conversion between polar and Cartesian (“normal” coordinates) system is \( x = r \cos \theta \) and \( y = r \sin \theta \). This makes converting graphs between systems possible, allowing a deeper understanding of their properties and behaviors.