Problem 4 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r=2 \sin \theta\)

Step-by-Step Solution

Answer

The graph of \(r = 2 \sin \theta\) is a circle centered at (0, 1) with a radius of 1.

Step 1: Understand the given equation

The given equation is in polar coordinates: \[ r = 2 \sin \theta \] Here, \( r \) is the radius (distance from the origin) and \( \theta \) is the angle.

Step 2: Identify key points

Identify angles \( \theta \) where \( \sin \theta \) has known values: - When \( \theta = 0 \), \( \sin 0 = 0 \ \rightarrow r = 0 \) - When \( \theta = \ \frac{\pi}{2} \), \( \sin \frac{\pi}{2} = 1 \ \rightarrow r = 2 \) - When \( \theta = \pi \), \( \sin \pi = 0 \ \rightarrow r = 0 \) - When \( \theta = \frac{3\pi}{2} \), \( \sin \frac{3\pi}{2} = -1 \ \rightarrow r = -2 \), which means \( r = 2 \) in the opposite direction (\( \theta + \pi \)).

Step 3: Plot the points

Using the identified points: 1. At \( \theta = 0 \), plot point (0, 0). 2. At \( \theta = \frac{\pi}{2} \), plot point (0, 2). 3. At \( \theta = \pi \), plot point (0, 0). 4. At \( \theta = \frac{3\pi}{2} \), plot point (0, -2), but this translates to the point (0, 2) in the opposite direction.

Step 4: Connect the points smoothly

The given equation describes a circle. Connect the plotted points smoothly to form the circle. The circle is centered at (0, 1) with a radius of 1 in the Cartesian plane.

Key Concepts

Graphing Polar Equations

Polar coordinates are a way to express points on a plane where each point is determined by a distance from the origin (the radius, denoted as \( r \)) and an angle from the positive x-axis (denoted as \( \theta \)). In the given exercise, we have the polar equation \( r = 2 \sin \theta \). When graphing polar equations, follow these steps:

• Identify key angles and calculate corresponding \( r \) values.
• Plot the points on the polar grid.
• Connect the points smoothly to form the graph.

In our example, we identified angles where the sine function is well-known, helping plot points accurately. Remember, the beauty of polar graphs is that they often form shapes like circles, spirals, or roses just by plotting points and connecting them.

Sinusoidal Functions

Sinusoidal functions, like sine and cosine, are fundamental in understanding polar equations. They describe oscillating patterns and are pivotal in creating smooth curves. For the equation \( r = 2 \sin \theta \), the radius \( r \) varies according to the sine of the angle, \( \theta \).
**Here's a quick overview of the sine function:**

• \( \sin 0 = 0 \)
• \( \sin \frac{\pi}{2} = 1 \)
• \( \sin \pi = 0 \)
• \( \sin \frac{3\pi}{2} = -1 \)

Notice how the sine function returns to zero or hits a maximum/minimum as the angle progresses. This cyclical behavior helps form predictable and often symmetric graphs in polar coordinates.

Plotting Points

Plotting points correctly in polar coordinates requires understanding both the distance from the origin (\( r \)) and the angle (\( \theta \)). Follow these guidelines for plotting:

• Identify specific angles where the function's values are known.
• Calculate the corresponding radius values.
• Mark the points on the polar grid at the calculated radii and angles.
• Consider the full rotation from 0 to \( 2\pi \) for a complete graph.

For \( r = 2 \sin \theta \), we used points like (0, 0), (\( \frac{\pi}{2} \), 2), (\( \pi \), 0), and (\( \frac{3\pi}{2} \), -2). Since \( r = -2 \) at \( \frac{3\pi}{2} \) translates to the opposite direction, it adds symmetry to our graph. Connect these plotted points smoothly to reveal the complete shape, often showing beautiful symmetries.