Problem 73 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r=2 \theta $$

Step-by-Step Solution

Answer

The graph of the polar equation \( r=2 \theta \) forms a spiral shape starting from the origin, spiralling counter-clockwise as the angle \( \theta \) increases. The overall form is a spiral going out from the origin of the graph. Distance drawn from origin increases linearly as we move around the angle \( \theta \).

Step 1: Understand Polar Coordinates

The polar coordinate system is 2-dimensional with each point expressed as a distance (r) and an angle (\( \theta \)) from a reference point and line. \( r=2 \theta \) indicates that the radial distance from the origin increases linearly as the angle \( \theta \) grows.

Step 2: Set up the Graph

To begin, it's recommended to set up a polar grid, which consists of concentric circles (representing the radial distance), and lines radiating out from the center of these circles (representing angles).

Step 3: Plot Points

To visualize the graph, start by taking sample points. Start with \( \theta = 0 \), which gives \( r = 0 \). This indicates the graph starts at the origin. Then try 90 degrees, 180 degrees, 270 degrees, and 360 degrees. Remember, the angle \( \theta \) is usually measured in radians in mathematics. Thus, instead of 90, 180, 270, and 360 degrees, you would likely be more accurate to try \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \) radians, respectively.

Step 4: Draw the Graph

Once you have a sufficient amount of points, begin to connect them. You should notice that as you increase in your \( \theta \) values, your points are spiraling outward from the origin, thus forming a spiral-like shape. Ensuring smooth connection between these dots will yield a graph for the given polar equation.