Question

Identify and graph each polar equation. $$ r=2 \sin (3 \theta) $$

Short answer

The equation \( r = 2 \, \text{sin}(3\theta) \) represents a rose curve with 3 petals, each extending 2 units from the origin.

Step-by-step solution

Understand the Polar Equation

The given polar equation is \( r = 2 \, \text{sin}(3\theta) \). In polar coordinates, \(r\) represents the radius and \(\theta\) represents the angle.

Identify the Type of Graph

The equation \( r = 2 \, \text{sin}(3\theta) \) is a polar equation that represents a rose curve. A rose curve with equation \( r = a \, \text{sin}(n\theta) \) has \(n\) petals if \(n\) is odd, and \(2n\) petals if \(n\) is even. Since \(n = 3\) here (which is odd), the graph will have 3 petals.

Determine Range and Key Angles

To graph the equation, calculate the values of \(r\) at key angles. Since the sine function completes a period at \(2\pi\), consider angles from \(0\) to \(2\pi\) at intervals like \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), etc.

Compute \(r\) for Selected Angles

Compute \(r\) values for these angles: \( \theta = 0 \Rightarrow r = 2 \, \text{sin}(0) = 0 \)\( \theta = \frac{\pi}{6} \Rightarrow r = 2 \, \text{sin}(\frac{\pi}{2}) = 2 \)\( \theta = \frac{\pi}{3} \Rightarrow r = 2 \, \text{sin}(\pi) = 0 \)\( \theta = \frac{\pi}{2} \Rightarrow r = 2 \, \text{sin}(\frac{3\pi}{2}) = -2 \)\( \theta = \frac{2\pi}{3} \Rightarrow r = 2 \, \text{sin}(2\pi) = 0 \)

Plot and Connect the Points

Using the calculated \(r\) values, plot the points on polar coordinate paper. Then, connect these points smoothly to form the rose curve with 3 petals. Each petal of the rose curve extends to a maximum distance of 2 units from the origin.

Key concepts

polar coordinates

Polar coordinates are a method of representing points on a plane using a radius and an angle. Unlike Cartesian coordinates, which use horizontal (x) and vertical (y) distances, polar coordinates specify the distance from a central point (usually called the origin) and the angle measured from a fixed direction (usually the positive x-axis).

The radius, denoted as \(r\), tells you how far the point is from the origin. The angle, denoted as \(\theta\), tells you the direction in which to move from the origin. When graphing polar equations, you represent each point by moving \(r\) units away from the origin at an angle of \(\theta\) radians or degrees.

rose curve

A rose curve is a type of graph that forms a flower-like shape with multiple petals. It's derived from polar equations of the form \( r = a \sin(n\theta) \) or \( r = a \cos(n\theta) \), where \(a\) and \(n\) are constants. The key to understanding rose curves is the number of petals they have:

• When \(n\) is odd, the curve will have \(n\) petals.
• When \(n\) is even, the curve will have \(2n\) petals.

For example, in the equation \(r = 2 \sin(3\theta)\), \(n\) is 3 (odd), so the graph will have 3 petals. The constant 2 indicates that each petal will extend to a maximum distance of 2 units from the origin.

graphing polar equations

Graphing polar equations requires a systematic approach. Follow these steps to create a clear graph:

1. Identify the type of polar equation you are dealing with.
2. Determine the key values of \(r\) at important angles like 0, \(\frac{\pi}{2}\), \(\pi\), etc.
3. Calculate \(r\) for these angles to get a set of points.
4. Plot these points on polar graph paper, which is designed with circles indicating different radii and lines showing different angles.
5. Connect the points smoothly, keeping in mind the symmetry and periodicity of the function.

This method helps in visualizing the complete shape of the curve accurately.

sine function

The sine function, denoted by \(\sin\), is a periodic function that oscillates between 1 and -1. It plays a crucial role in polar equations like \(r = 2 \sin(3\theta)\). Key properties of the sine function include:

• It completes one full wave from 0 to \(2\pi\) radians.
• It is zero at angles like 0, \(\pi\), \(2\pi\), etc.
• It reaches its maximum value of 1 at \(\frac{\pi}{2}\), and its minimum value of -1 at \(\frac{3\pi}{2}\).

These properties help in understanding the periodic nature of the polar equation and in determining the shape and extent of the curve's petals.

angle calculation

Angle calculation is a fundamental step in graphing polar equations. To get a complete picture of the graph, you need to evaluate the radius \(r\) at several key angles covering one full period (0 to \(2\pi\)).

Here are some examples based on the equation \(r = 2 \sin(3\theta)\):

• \(\theta = 0\)
• \(\theta = \frac{\pi}{6}\)
• \(\theta = \frac{\pi}{3}\)
• \(\theta = \frac{\pi}{2}\)
• \(\theta = \frac{2\pi}{3}\)

For each angle, substitute \(\theta\) into the equation to solve for \(r\). These calculations provide the exact points to plot on your polar graph, ensuring accuracy in depicting the rose curve. This process also highlights how the graph extends and folds back, forming its characteristic petals.