Problem 72 - 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. $$ r=1+\cos \theta ; r=3 \cos \theta $$

Step-by-Step Solution

Answer

Points of intersection: \( \left( \frac{3}{2}, \frac{\pi}{3} \right) \) and \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \).

Step 1 - Understand the Polar Equations

The given polar equations are: \[ r = 1 + \cos \theta \]\[ r = 3 \cos \theta \].In these equations, \( r \) is the radius and \( \theta \) is the angle in the polar coordinate system.

Step 2 - Set the Equations Equal to Each Other

To find the points of intersection, set the two equations equal to each other:\[ 1 + \cos \theta = 3 \cos \theta \].

Step 3 - Solve for \(\cos \theta\)

Rearrange the equation to solve for \( \cos \theta \):\[ 1 + \cos \theta = 3 \cos \theta \]\[ 1 = 2 \cos \theta \]\[ \cos \theta = \frac{1}{2} \].This happens when \( \theta \) is either \( \frac{\pi}{3} \) or \( \frac{5\pi}{3} \).

Step 4 - Plug \( \theta \) Back into One of the Equations

Now substitute \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) back into either of the original equations to find the corresponding \( r \) values:\[ r = 3 \cos \frac{\pi}{3} = 3 \times \frac{1}{2} = \frac{3}{2} \].Therefore, the points of intersection in polar coordinates are \( \left( \frac{3}{2}, \frac{\pi}{3} \right) \) and \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \).

Step 5 - Graph the Polar Equations

Graph both polar equations on the same polar grid. The first equation, \( r = 1 + \cos \theta \), is a limaçon with a dimple, and the second equation, \( r = 3 \cos \theta \), is a circle.

Step 6 - Label the Points of Intersection

On the graph, identify and label the points of intersection: \( \left( \frac{3}{2}, \frac{\pi}{3} \right) \) and \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \).

Key Concepts

Intersection of Polar Equations

To find the intersection points of two polar equations, you need to set the equations equal to each other. In this example, the given equations are:\r

\r
• \r\( r = 1 + \cos \theta \)\r
\r
• \r\( r = 3 \cos \theta \)\r
\r

\rBy setting them equal: \( 1 + \cos \theta = 3 \cos \theta \), solve for \( \cos \theta \) first. Then, you plug the value of \( \cos \theta \) into one of the original equations to find the corresponding \( r \) values. These coordinates are the points of intersection. For this specific problem, setting the equations equal leads to:\r\( \cos \theta = \frac{1}{2} \), which gives \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \). Substituting these into the equations, we get the points \( \left( \frac{3}{2}, \frac{\pi}{3} \right) \) and \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \).\r

Graphing Polar Coordinates

When graphing polar coordinates, the position of each point is determined by a radius \( r \) and an angle \( \theta \). Use a polar grid, which is a circular graph divided into angles and radii. In our exercise, we graph the equations:\r

\r
• \r\( r = 1 + \cos \theta \)\r
\r
• \r\( r = 3 \cos \theta \)\r
\r

\rThe first equation, \( r = 1 + \cos \theta \), forms a limaçon with a dimple, a distinct shape that curves inward. The second equation, \( r = 3 \cos \theta \), forms a circle with radius 3 centered along the polar axis. By plotting these equations on the same grid, you can visually identify their points of intersection.

Solving Trigonometric Equations

To solve trigonometric equations, you'll often use fundamental identities and algebraic manipulations. Here, we started with:\r\( 1 + \cos \theta = 3 \cos \theta \).\rRearrange it to:\r\( 1 = 2 \cos \theta \),\rthen isolate \( \cos \theta \):\r\( \cos \theta = \frac{1}{2} \).\rFrom this, we know \( \theta \) values from standard trigonometric values. The solutions are at angles where \( \cos \theta = \frac{1}{2} \), leading to \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).\rAfter finding \( \theta \), substitute these back into one of the original equations to find \( r \). Understanding these steps can simplify complex problems and help understand the behavior of the functions.