Problem 27 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=\sin \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r=\sin \theta\) is a circle of radius 0.5 centered at (0,0.5) in the Cartesian plane, respecting the symmetry of the graph with respect to the x-axis.

Step 1: Identify the Zeros

To find the zeros of \(r\), set \(r = \sin \theta = 0\). The solutions to this equation are \(\theta = 0, \pi\), indicating that the graph intersects the polar axis at these points.

Step 2: Identify the Maximum Values

To find the maximum values of \(r\), look for the maximum value of \(\sin \theta\), which is 1. Therefore, the maximum value of \(r\) is 1 when \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\). This gives two points on the graph: \((r, \theta) = (1, \frac{\pi}{2}), (1, \frac{3\pi}{2})\).

Step 3: Identify the Symmetry

The graph has symmetry with respect to the x-axis because \(\sin(-\theta) = -\sin(\theta)\). Thus, the graph is symmetric about the y-axis in the polar coordinate plane.

Step 4: Sketch the Graph

Plot the points found and draw a smooth curve that passes through them while respecting the symmetry of the graph. The graph is a circle of radius 0.5 located at 0.5 above the origin in the polar plane.

Key Concepts

Graphing Polar Coordinates

Polar coordinates describe positions on a plane based on angle and distance. Unlike the Cartesian system, which uses horizontal and vertical positioning, polar coordinates use a point known as the pole (origin) and an angle that indicates direction from a reference line known as the polar axis. The unit of the angle is typically radians.
When graphing polar equations, consider the following:

• Determine the behavior of the function. Check periodicity and range of values for defining critical points like zeros and maximums.
• Identify crucial angles. These reference points will help you understand how the graph is structured as these values often repeat across certain span of angles.

Using our example equation, \(r = \sin \theta\), it's evident the graph is derived from the sinusoidal functions' cyclic nature. Start plotting by examining the behavior of \(r\) as \(\theta\) progresses from 0 to \(2\pi\). This will allow you to better map out how the graph should look visually.

Maximum Values in Polar Graphs

When studying polar graphs, identifying points where the radius \(r\) achieves its largest value is key. These maximums often help define the graph’s shape and extent.
For the equation \(r = \sin \theta\), the maximum value of \(r\) occurs when \(\sin \theta = 1\). Therefore, the maximum value of \(r\) is 1. This happens at angles where \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).
These points represent the furthest distance the graph extends from the origin. By understanding this maximum, you can plot how large the graph will be, which is crucial for sketching its overall form accurately.

• Mark the points corresponding with maximum radius directly on the graph.
• Repeat maximum markings for all periodic recurrences of such values within one cycle of the angle, \(\theta\), ensuring completeness.

For this equation, the result is a graph resembling a circle at a particular height, arising due to the nature of \(\sin \theta\) peaking at these intervals.

Symmetry in Polar Graphs

Symmetry in polar graphs helps simplify graphing efforts and confirms that certain predicted symmetric forms do occur.
One type of symmetry, for example, is reflection symmetry across one of the axes, which is quite common in polar equations.
For \(r = \sin \theta\), the graph shows reflection symmetry across the y-axis (vertical line symmetry) because \(\sin (-\theta) = -\sin (\theta)\). This means any point (\(r, \theta\)) will have a counterpart (-\(r, -\theta\)), which engineers what's known as polar symmetry.
Understanding symmetry allows simplifying curve sketching, as you would effectively only need to draw a part of the curve and reflect across the line of symmetry:

• Determine the types of symmetry present: axis symmetry or point symmetry.
• Mirror plotted points across the symmetry line for full visualization.

By mastering symmetry concepts, sketching the entire polar graph becomes efficient and highly accurate.