Problem 16 - Essential Calculus

Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=1+\cos \theta \\ r=3 \cos \theta \end{array} $$

Step-by-Step Solution

Answer

The points of intersection of the graphs of the given equations are \((1.5, \pi/3)\) and \((1.5, 5\pi/3)\) in polar form.

Step 1: Equate the Two Equations

To find the points of intersection, both equations should be set equal to each other. Thus, write the equation as \(1 + \cos \theta = 3\cos \theta\).

Step 2: Solve for Cosine Theta

The next step is to isolate \(\cos \theta\) on one side. So, subtract \( \cos \theta\) from both sides of the equation, resulting in \(1 = 2\cos \theta\). Then, divide both sides by 2 to isolate \(\cos \theta\). Therefore, \(\cos \theta = 1/2\).

Step 3: Solve for Theta

Knowing that the cosine of an angle is 1/2, find the value of \(\theta\). In the unit circle, \(\cos \theta = 1/2\) at \(\theta = \pi/3\) and \(\theta = 5\pi/3\). So our solutions for \(\theta\) are \(\pi/3\) and \(5\pi/3\).

Step 4: Substitute Theta Values in Original Equations

Substitute the \(\theta\) values found in both the original equations to find the corresponding \(r\) values. Substituting \(\theta = \pi/3\) in \(r=3\cos\theta\), we get \(r = 3*\cos(\pi/3)\) which simplifies to \(r = 3*(1/2) = 1.5\). Similarly, substituting \(\theta = 5\pi/3\) in \(r=3\cos\theta\), we get \(r = 3*\cos(5\pi/3)\) which simplifies to \(r = 3*(1/2) = 1.5\) too. So, the coordinate points in polar form for the intersections are \((1.5, \pi/3)\) and \((1.5, 5\pi/3)\).

Key Concepts

Points of Intersection

Finding the points of intersection between two polar equations involves a direct comparison of their expressions. Unlike the Cartesian coordinate system where you might solve for \(x\) and \(y\), in polar coordinates, you work with the radial distance \(r\) and the angle \(\theta\). To find where two curves intersect, you equate the expressions for \(r\) from both polar equations.
Consider an example where we have two equations: \r=1+\cos \theta\ and \r=3\cos \theta\. By setting them equal to each other, you obtain \(1 + \cos \theta = 3\cos \theta\). This allows you to solve for the angle \(\theta\) at which the intersection occurs. Such intersections are crucial for understanding the relationship and overlap between two curves.

To solve these problems effectively, focus on simplifying the equation as much as possible, which often involves reducing terms and isolating one of the trigonometric functions. Once \(\theta\) values are identified, substitute these back into any of the original equations to find \(r\), thus obtaining full polar coordinates of intersection points.

Trigonometric Equations

Trigonometric equations in polar coordinates can appear daunting, but they often boil down to familiar trigonometric values. In our example, the task was to understand when \(\cos \theta = \frac{1}{2}\). This is a classic trigonometric value.
To solve trigonometric equations, you should recall unit circle values where these functions take common values, like \sin, \cos, ext{ and } \tan\ at specific angles. The \(\cos \theta = \frac{1}{2}\) equation points out that the angle \(\theta\) could be \frac{\pi}{3}\ or \frac{5\pi}{3}\, given the symmetry of trigonometric functions.

These equations often equate the trigonometric function to basic values, and by referring to standard angles from the unit circle, you can identify potential solutions for \(\theta\). Look always for principal angles first; they provide a solid start for solving such problems.

Unit Circle

The unit circle is a key concept in trigonometry that aids in solving polar coordinate problems. It's a circle of radius one, centered at the origin of the coordinate plane, and provides the framework for identifying trigonometric function values at various angles \(\theta\).
In polar coordinates, the angle \(\theta\) corresponds to positions on the unit circle. For example, on the unit circle, the point where \(\cos \theta = \frac{1}{2}\) can be matched with both \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\) angles. These are periodic points where the same trigonometric values recur, demonstrating the periodic nature of sine and cosine functions.

By familiarizing yourself with the unit circle, you can quickly resolve trigonometric equations encountered in polar coordinates by matching angles to their known trigonometric ratios. Memorize common angles and their sine and cosine values to make these problems more manageable.

Solving Equations

Solving equations in polar coordinates involves a logical approach to manage both \(r\) and \(\theta\) components effectively. As seen in the given problem, setting the equations equal allows you to simplify and solve for a single variable. Start by isolating the trigonometric function concerned—in this case, \(\cos \theta\).
The ability to rearrange terms, factor common values, or divide to isolate a variable is essential in simplifying polar coordinate equations, much like in linear equations. After finding the values of the angle \(\theta\),

• verify by plugging these back into one of the original equations to find \(r\);
• cross-check the results using both equations to confirm that \(r\) values make sense for each \(\theta\);
• restate the solution clearly in terms of polar coordinates, reflecting your findings in an \(r, \theta\) format.

Mastering these steps can drastically improve accuracy and understanding when dealing with similar problems. Efficient equation manipulation, coupled with strong trigonometric insight, results in grasping intersections in polar graphs.