Problem 30 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=4(1-\sin \theta)$$

Step-by-Step Solution

Answer

The graph of the equation \(r=4(1-\sin \theta)\) is a circle centered at (0, 2) with a radius of 2, passing through the pole at \(\theta = \pi/2\) and having its maximum distance from the pole (8 units) at \(\theta = 3\pi/2\).

Step 1: Test for Symmetry

To determine symmetry around the x-axis, substitute \(-\theta\) for \(\theta\). If the equation is unchanged, then it’s symmetric with respect to the x-axis. Similarly, by substitifying \(\pi - \theta\) for \(\theta\), we can test the symmetry around the y-axis. If it maintains the same equation, it is symmetric with respect to the y-axis. Lastly, substituting \(\pi + \theta\) for \(\theta\), we can find the symmetry with respect to the origin. Here, on substitution, the equation remains same, thus it is symmetric with respect to the Polar axis (not x, y or origin).

Step 2: Finding the Zeros

We find the zeros by setting \(r\) equal to zero and solving for \(\theta\). In this case, when \(r = 4(1 - \sin \theta) = 0\), solving gives \(\sin \theta = 1\). Thus, the zeros are at \(\theta = \frac{\pi}{2} + 2n\pi\),

Step 3: Determining Maximum \(r\)-values

The maximum \(r\)-values are the results of the maximum values of \(4(1-\sin \theta)\). This occurs at \(\sin \theta\) minimum which is -1 and results \(r\) = 4(1-(-1)) = 8. Thus, the maximum \(r\)-value is 8.

Step 4: Add Other Additional Points

To make sure enough points are available to accurately sketch the graph, we need to solve the equation for more points. Here are some point pairs (\(\theta\), \(r\)): (0, 4), (\(\pi/2\), 0), (\(\pi\), 4), and (3\(\pi/2\), 8).

Step 5: Sketch the Graph

Plot the points and join them to get the graph. The graph turns out to be a circle with a center at (0, 2) and radius 2. The circle passes through the pole at \(\theta = \pi/2\) and has its maximum distance from the pole (8 units) at \(\theta = 3\pi/2\).

Key Concepts

Polar Equations

Polar equations express relationships between the radial distance from the origin, denoted as \( r \), and the angle \( \theta \), measured in radians from the positive x-axis in the polar coordinate system. These equations can often represent complex curves and shapes, such as circles, spirals, and roses, which are more intricate to express in Cartesian coordinates.

For instance, the polar equation \( r = 4(1 - \sin \theta) \) describes a circle with a center that is not at the origin. By manipulating this equation, we can locate key features of the graph, including its symmetry, zeros, and maximum radial distances, which help when sketching the curve. The equation's form suggests the presence of a sinusoidal component, which we'll discuss further in the context of sinusoidal functions.

Sinusoidal Functions

Sinusoidal functions are mathematical functions that describe smooth, periodic oscillations. They are typically represented as \( \sin \theta \) or \( \cos \theta \) and are fundamental to understanding waves and harmonic motion. In polar equations, sinusoids often determine the radial distance as a function of angle, leading to shapes that exhibit cyclical patterns.

In our example, \( r = 4(1 - \sin \theta) \), the sinusoidal function \( \sin \theta \) transforms a basic circle into one with a varying radius as \( \theta \) changes. The amplitude and phase shift of the sinusoidal function dictate the size and orientation of the loop or petal-shaped features in polar graphs. Understanding sinusoidal functions is crucial when interpreting these varying radii in polar graphs.

Graph Symmetry

Graph symmetry in polar coordinates can dramatically simplify the process of sketching graphs. Recognizing symmetry helps reduce the amount of computation needed, as symmetric points can be inferred rather than calculated. A graph can exhibit three types of symmetry:

• Symmetry with respect to the line \( \theta = 0 \), also known as polar axis symmetry.
• Symmetry with respect to the line \( \theta = \frac{\pi}{2} \), perpendicular to the polar axis.
• Symmetry with respect to the pole (origin).

For the equation \( r = 4(1 - \sin \theta) \), by substituting \( -\theta \), \( \pi - \theta \), and \( \pi + \theta \) in place of \( \theta \), we only find that the original equation remains unchanged for \( \pi + \theta \). Therefore, the graph exhibits polar axis symmetry. This insight saves time by allowing us to reflect points across the polar axis rather than calculating each independently.

Polar Coordinates

Polar coordinates define the location of a point based on its angle and distance from a central point (the pole). Unlike Cartesian coordinates which use a grid of x and y values, polar coordinates use the notation \( (r, \theta) \), where \( r \) is the radial distance from the pole, and \( \theta \) is the angle from the positive x-axis.

A vital application of polar coordinates is in representing curves that are cumbersome to express with x and y coordinates. Understanding how to plot points given in polar form is essential for sketching polar graphs. It's also necessary for students to understand how polar equations can be converted and compared to their Cartesian equivalents for a deeper comprehension of the correspondence between these two coordinate systems.