Problem 19 - Essential Calculus

Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=4 \sin 2 \theta \\ r=2 \end{array} $$

Step-by-Step Solution

Answer

The points of intersection of the graphs of the equations \(r = 4 \sin(2\theta)\) and \(r = 2\) are \((2, \frac{\pi}{12})\) and \((2, \frac{5\pi}{12})\).

Step 1: Set the Two Polar Equations Equal

To find the intersections of the graphs of the two equations, we set them equal to each other, resulting in: \[4 \sin(2\theta) = 2\] or \[\sin(2\theta) = \frac{1}{2}\]

Step 2: Solve for Theta

Solving the equation \(\sin(2\theta) = \frac{1}{2}\), we get: \[2\theta = \frac{\pi}{6}, \frac{5\pi}{6}\] then we divide these results by 2, getting: \[\theta = \frac{\pi}{12}, \frac{5\pi}{12}\]

Step 3: Determine the Intersection Points

Next, we substitute the values of \(\theta\) into one of our original equations (it doesn't matter which one since they're equal at the points of intersection). Substituting into \(r = 2\) yields the points of intersection: \[(2, \frac{\pi}{12}), (2, \frac{5\pi}{12})\]

Key Concepts

Points of Intersection

When dealing with graphs, finding points of intersection is like playing detective. It involves identifying where two graphs meet or cross each other on a coordinate plane. Imagine two roads that intersect; the point where they cross is analogous to a point of intersection in graph terms. In polar coordinates, these points are determined by the pair \(r, \theta\), where \(r\) is the radius (or distance from the origin) and \(\theta\) is the angle.

For our given problem, we need to find where the graphs of both polar equations, \(r = 4 \sin 2\theta\) and \(r = 2\), intersect. To discover this, we set the 'r' values from both equations equal to each other and solve for \(\theta\). By doing so, we're pinpointing the exact locations in the polar coordinate system where the graphs meet.

Polar Equations

Polar equations represent graphs using a different coordinate system than Cartesian coordinates. Instead of using x and y axes, polar coordinates use values of \((r, \theta)\).

Here's what you need to know about polar coordinates:

• \(r\): represents the distance from the origin (pole).
• \(\theta\): represents the angle in radians from the positive x-axis, measured counterclockwise.

For the equations \(r = 4 \sin 2\theta\) and \(r = 2\), the first equation is dependent on the angle \(\theta\), showcasing a dynamic curve often forming intricate shapes like roses. The second equation is a circle with a radius of 2 centered at the origin. By examining these unique equations, understanding their nature and how changing \(\theta\) affects them, you're better equipped to analyze and solve these types of mathematical challenges.

Solving Trigonometric Equations

Solving trigonometric equations involves finding angles that satisfy given conditions, often relying on relationships between angles and sides in triangles. In the given problem, we look to solve \(\sin(2\theta) = \frac{1}{2}\).

Here's how you can break it down:

• First, identify angles where sine equates to \(\frac{1}{2}\), typically \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\) in a standard trigonometric circle.
• However, since the equation is \(\sin(2\theta)\), we address these values as \(2\theta\), not \(\theta\) itself.
• After finding \(2\theta\), we divide by 2 to isolate \(\theta\), yielding \(\frac{\pi}{12}\) and \(\frac{5\pi}{12}\).

Once you determine these values for \(\theta\), substituting them back into one of the original polar equations confirms their validity as intersection points, leading to a complete solution.