Problem 46 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=2-3 \sin \theta$$

Step-by-Step Solution

Answer

To solve this problem, value pairs of \( (r,\theta) \) were computed and plotted on a polar graph. The resulting shape is a 'loop' or a 'petal'.

Step 1: Create a table with values of theta

First, you need to generate a table with values of \( \theta \) starting from \( \theta = 0 \) to \( \theta = 2\pi \). You can choose the step size based on the precision you want. Here, we'll take increments of \( \pi/2 \) to keep things simple.

Step 2: Plug the values of \( \theta \)

Now replace each value of \( \theta \) in the sine function \(r=2-3 \sin \theta\). This results to an equivalent \( r \) value. By doing so, you obtain the polar coordinates \((r, \theta)\).

Step 3: Plot the polar coordinates

After obtaining the polar coordinates, you can plot them on polar graph paper. This involves marking an angle \( \theta \), then moving away from the origin by distance \( r \). If \( r \) is negative, this means moving in the opposite direction of \( \theta \).

Step 4: Connect the points

Finally, join the plotted points to create the graph of the given polar equation. The shape of the graph should resemble a 'loop' or a 'petal'.

Key Concepts

Polar Coordinates

When it comes to understanding the location of a point in a plane, we're often accustomed to the rectangular coordinate system, which uses horizontal and vertical axes. But there's another way: polar coordinates. To grasp its essence, imagine a point's position defined not by its x and y values, but instead by how far away and at what angle it is from a central point, known as the pole.

Polar coordinates come as pairs \( (r, \theta) \) where \( r \) represents the radius - the distance from the pole - and \( \theta \) is the angle measured from the positive x-axis, often called the polar axis. Positive \( r \) values indicate that the point lies in the direction of \( \theta \) from the pole, while negative values mean it's in the exact opposite direction. This can initially be counterintuitive, but with practice, defining a point's location becomes second nature.

Examples Simplified

To make it more digestible, let's say you're telling a friend where a picnic spot is in a park. Instead of giving a traditional address (rectangular coordinates), you might say, 'It's 100 meters away in the direction of the big oak tree (angle).' That's the essence of polar coordinates - direction and distance.

Polar Equations

Polar equations are the rules that describe a relationship between the radius \( r \) and the angle \( \theta \) in polar coordinates. Instead of the familiar y = mx + b linear equation from algebra, polar equations can take on a variety of forms such as \( r = a + b\sin(\theta) \) or \( r = a\cos(\theta) \) just to name a couple. These equations often result in elegant, intricate graph patterns like spirals, circles, and petals, depending on the constants and functions involved.

One could see these equations as the formulas behind the beauty of the rose or the spiral pattern of the galaxies. They're that fundamental.

Interpreting the Equation

Understanding the meaning of a polar equation involves recognizing how the values of \( r \) change as \( \theta \) increases. For the equation \( r = 2 - 3\sin(\theta) \) that we are considering here, the value of \( r \) fluctuates based on the sine of \( \theta \) with a baseline at 2 and an amplitude of -3. The negative sign indicates that our graph will have some interesting features, such as loops or petals.

Graphing in Polar Systems

To visualize a polar equation, one needs to graph it in a setup designed for polar coordinates. Unlike Cartesian graphs with horizontal and vertical lines as grids, polar graphs feature circles and rays emanating from the center point, allowing us to plot the coordinates \( (r, \theta) \) more intuitively.

The initial step usually involves calculating several pairs of polar coordinates by plugging various \( \theta \) values into the polar equation, just as we saw in the textbook exercise. Then, these coordinates are plotted on the polar graph. Remember, for positive \( r \) values, the point is plotted along the direction of \( \theta \), and for negative \( r \) values, in the opposite direction.

Polar Graph Features

Some prominent features of these graphs are circles, spirals, and rose curves—each formation giving a clue about the underlying equation. For example, if an equation contains a sinusoidal function like sine or cosine, expect to see symmetrical patterns with loops. Graphing in polar systems highlights the symmetrical nature and the repetitive patterns of many mathematical phenomena – from the petals of a flower to the orbits of celestial bodies.