Problem 38 - Precalculus

Question

Graph equation. \(r=3 \sin \theta\)

Step-by-Step Solution

Answer

The graph of \(r=3 \sin \theta\) is a circle centered at \((0, \frac{3}{2})\) with a diameter of 3.

Step 1: Understand the Equation Type

The equation given is in polar coordinates, which are represented in the form of \( r = f(\theta) \). In this case, \(r = 3 \sin \theta\). This describes a curve where the radius \(r\) changes depending on the angle \(\theta\).

Step 2: Recall the Shape for Sine-based Polar Equations

For equations of the form \(r = a \sin \theta\), the graph is typically a circle. Specifically, when \(a\) is positive, the circle is symmetric about the vertical axis (the y-axis in Cartesian coordinates). Here, \(a\ = 3\). This indicates the presence of a circle.

Step 3: Determine the Circle's Characteristics

Given \(r = 3 \sin \theta\), identify the maximum value of \(\sin \theta\), which is \(1\). Thus, the maximum radius \(r\) is 3. The minimum when \(\sin \theta = 0\) is \(r = 0\). This confirms that the circle's diameter is 3 and it is centered at \((0, \frac{3}{2})\) in Cartesian coordinates.

Step 4: Sketch the Graph

Draw a circle centered at \((0, \frac{3}{2})\) on the polar plane. Since the function involves \(\sin\), the circle is oriented vertically. The radius being \(3/2\) means the circle will stretch from the pole (origin) to a full extent vertically along the initial line (y-axis). Its diameter is vertically positioned from \(\theta = \frac{\pi}{2}\) to \(-\frac{\pi}{2}\).

Step 5: Verify Symmetry and Features

Double-check the shape: It should be circular, symmetric about the vertical axis. This confirms our derivation: The circle's radius extends up to 3 along the vertical axis, and passes through the origin with its bottom-most point at \((0, 0)\) when \(\theta = 0\ or \theta = \pi\).

Key Concepts

Sine Function

The sine function, represented as \( \sin \theta \), is a fundamental trigonometric function that oscillates between -1 and 1. In the context of polar coordinates, it plays a critical role in determining the radius \( r \) based on the angle \( \theta \). This is evident in the polar equation \( r = 3 \sin \theta \). Here, the sine function modifies the radius as \( \theta \) changes:

• When \( \sin \theta = 1 \), the radius \( r \) is at its maximum of 3.
• When \( \sin \theta = 0 \), the radius \( r \) is zero.

Understanding the sine function's properties aids in predicting how the graph will behave. Particularly, it helps identify that the values of \( r \) smoothly transition from zero at \( \theta = 0 \), reach their peak at \( \theta = \frac{\pi}{2} \), and decrease back to zero. This periodic nature helps us predict and sketch the curve accurately in the polar coordinate system.

Circle Graphing

In polar coordinates, graphing a circle is more straightforward if you know the equation's form. A sine-based polar equation like \( r = 3 \sin \theta \) typically results in a circular graph. To graph a circle from this equation:

• Identify that the function involves a sine term, indicating a circular shape oriented in a specific direction.
• Recognize that the maximum value of the equation determines the circle's diameter. Here, \( a = 3 \) means the diameter is 3.

This circular shape is centered along the y-axis in Cartesian coordinates, with its full extent being from the origin upwards. Since the sine function affects how \( r \) evolves as \( \theta \) changes, you can determine that the full shape of the circle is reached at \( \theta = \frac{\pi}{2} \). As a result, the circle is symmetric and vertically directed with its highest point extending to the maximum radius.

Symmetry in Polar Graphs

Symmetry is an essential aspect when analyzing and graphing polar equations. The given equation, \( r = 3 \sin \theta \), shows a specific symmetry, allowing for easier graph interpretation.

• Because the equation involves \( \sin \theta \) and the coefficient 3 is positive, the symmetry is about the vertical axis or y-axis.
• This line of symmetry simplifies the process of determining how \( r \) varies since you know the graph should mirror itself across this axis.

In practical terms, this symmetry means that once you've plotted half of the circle from \( \theta = -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), you don't need to re-calculate or re-graph for the rest of its rotation. The graph simply reflects, showing the same structure on either side of \( \theta = 0 \). This symmetry aids significantly in graphing and understanding polar plots with sine functions.