Problem 14 - Essential Calculus

Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=2-3 \cos \theta \\ r=\cos \theta \end{array} $$

Step-by-Step Solution

Answer

The points of intersection of the polar curves are \((\frac{1}{2},\frac{\pi}{3})\) and \((\frac{1}{2},\frac{5\pi}{3})\).

Step 1: Equate the two given equations

Set \(2 - 3 \cos \theta = \cos \theta \). This equation will represent the \(\theta \) where the two polar curves intersect.

Step 2: Solve for \(\theta \)

By rearranging the equation in Step 1, we get \(4 \cos \theta = 2 \), which simplifies to \(\cos \theta = \frac{1}{2}\). By referencing a unit circle or a trigonometric table, we can confirm that \(\theta = \frac{\pi}{3}\) or \(\theta = -\frac{\pi}{3} \). However, because we are dealing with a cos function which has a positive period of \(2\pi \), \( -\frac{\pi}{3}\) can also be represented as \( \frac{5\pi}{3} \). So the valid intersection points are \(\theta = \frac{\pi}{3}\) and \( \theta = \frac{5\pi}{3}\)

Step 3: Find the intersection points

Now, in polar coordinates we also need the radial coordinate \(r\). We substitute \(\theta\) in either of the initial equations to find \(r\). Using the second equation \(r=\cos \theta \), we can see that when \(\theta = \frac{\pi}{3}\) or \( \theta = \frac{5\pi}{3}\), \(r = \frac{1}{2}\). So, the points of intersection in polar coordinates are \((\frac{1}{2},\frac{\pi}{3})\) and \((\frac{1}{2},\frac{5\pi}{3})\).

Key Concepts

Trigonometric Solutions

Understanding trigonometric solutions is crucial when dealing with polar coordinates, as it often involves manipulating trigonometric equations. When solving for variable \( \theta \) in polar equations, we start by equating given equations. Here, we have two equations: \( r = 2 - 3 \cos \theta \) and \( r = \cos \theta \). The first step is to set these equal to each other: \( 2 - 3 \cos \theta = \cos \theta \).

To find \( \theta \), we rearrange and simplify this equation. Combining like terms, we get \( 4 \cos \theta = 2 \). Dividing by 4 gives us \( \cos \theta = \frac{1}{2} \). This is a classic trigonometric problem where we seek the corresponding angles \( \theta \).

Using a unit circle or trigonometric table, we find that \( \theta = \frac{\pi}{3} \) or \( \theta = -\frac{\pi}{3} \), adjusted for the full circle to \( \theta = \frac{5\pi}{3} \). This gives us the angles at which our polar graphs intersect. Understanding these trigonometric basics is key to successfully navigating problems involving angles and circular functions in polar coordinates.

Polar Equations

Polar equations use coordinates in a polar format, which differs vastly from Cartesian coordinates. In polar coordinates, each point on a plane is determined by a distance \( r \) from a reference point and an angle \( \theta \) from a reference direction. Our example uses polar equations \( r = 2 - 3 \cos \theta \) and \( r = \cos \theta \), which are examples of different forms that a polar equation can take.

The radial coordinate \( r \) represents the distance from the pole (origin), while the angle \( \theta \) represents the direction of the radius vector. It's different from Cartesian coordinates where points are denoted as \( (x, y) \). Polar equations often feature trigonometric functions like \( \sin \theta \) or \( \cos \theta \), which can define complex curves including circles, spirals, or roses.

When working with polar equations, it helps to understand how transforming or solving them provides insight into the curve's shape and position in polar coordinates. Each solution \( \theta \) offers new perspectives on the possible locations and characteristics of these curves.

Graph Intersection

Finding the intersection of polar graphs involves determining the exact points where two polar equations coincide. This exercise needs both solving and graphing skills, as we overlay the graphs of the given polar equations: \( r = 2 - 3 \cos \theta \) and \( r = \cos \theta \).

Identifying intersection points requires solving for \( \theta \) at which \( r \) values from both equations match. As detailed in the original solution, it's where \( \cos \theta = \frac{1}{2} \), leading to angles \( \theta = \frac{\pi}{3} \) or \( \theta = \frac{5\pi}{3} \).

Upon finding \( \theta \), substitute these back into either polar equation to calculate \( r \). For our exercise, substituting into \( r = \cos \theta \) gives \( r = \frac{1}{2} \). Hence, the intersection points are \( (\frac{1}{2}, \frac{\pi}{3}) \) and \( (\frac{1}{2}, \frac{5\pi}{3}) \). These intersections reveal where the plots of the polar equations overlap, aiding in graph interpretations and analyses.