Problem 70 - Algebra and Trigonometry

Question

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=3 ; r=2+2 \cos \theta $$

Step-by-Step Solution

Answer

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

Step 1 - Understand the Equations

The first polar equation given is a circle with radius 3: \[ r = 3 \]. The second polar equation is a limaçon: \[ r = 2 + 2 \cos \theta \].

Step 2 - Set the Equations Equal to Find Intersection Points

To find the points of intersection, set the equations equal to each other: \[ 3 = 2 + 2 \cos \theta \].

Step 3 - Solve for \theta

Subtract 2 from both sides: \[ 1 = 2 \cos \theta \]. Divide by 2: \[ \cos \theta = \frac{1}{2} \]. Solve for \theta to get: \[ \theta = \frac{\pi}{3}, \ \frac{5\pi}{3} \].

Step 4 - Find the Polar Coordinates

Substitute \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) back into either of the original equations to get \( r = 3 \). Thus, the points of intersection are: \( \left( 3, \frac{\pi}{3} \right) \) and \( \left( 3, \frac{5\pi}{3} \right) \).

Step 5 - Graph the Equations

Plot both equations on the same polar grid and mark the points of intersection. Label the points \( \left( 3, \frac{\pi}{3} \right) \) and \( \left( 3, \frac{5\pi}{3} \right) \) on the graph.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that describe a relationship between the radius \(r\) and the angle \(\theta\) in polar coordinates. Instead of using \(x\) and \(y\) coordinates like in Cartesian systems, polar equations use \(r\) to represent the distance from the origin and \(\theta\) for the angle from the positive x-axis. This can simplify the representation of curves, especially those that are naturally circular or spiral in nature.

Intersecting Polar Graphs

When graphing polar equations, it's often essential to find the points where two curves intersect. To determine these points, you set the two polar equations equal to each other and solve for \(\theta\).

Limaçon

A limaçon is a type of polar curve with general equation \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\). Its shape can vary significantly based on the values of \(a\) and \(b\), potentially featuring a dimple, loop, or appearing as a cardioid.

Solution for θ

To solve for \(\theta\) where polar equations intersect, set the equations equal to each other and solve for \(\cos \theta\) or \(\sin \theta\), then find \(\theta\) values that satisfy the equation.