Problem 80 - Precalculus with Limits

Question

Use a graphing utility to graph and identify \(r=2\ +\ k\sin\ \theta\) for \(k=0, 1, 2,\) and \(3\).

Step-by-Step Solution

Answer

The graphs created are all circles centered around r = 2, but as k increases, the influence of the sin function becomes more significant, increasing the size of the 'loops'. The loop's location moves further away from the origin as k increases.

Step 1: Graph for k = 0

When \(k=0\), the equation simplifies to \(r=2\). This represents a circle at distance 2 from the origin. Sketch this graph by plotting points along the circle of radius 2.

Step 2: Graph for k = 1

When \(k=1\), the equation is \(r=2+\sin\theta\). The \(\sin\) function modulates the radius, so the graph has a 'loop' where the radius goes to zero. The circle is now centered on \(r=2\), but includes a loop around the pole at (0,0) because the sin function varies between -1 and 1. Plot this graph's points on the same coordinate plane.

Step 3: Graph for k = 2

The equation becomes \(r=2+2\sin\theta\). Now the sin function has a greater influence. The resulting graph has a larger loop. The highest point on the loop will be when \(r = 4\) (at \(\theta = \pi/2\) or \(3\pi/2\)), and the lowest point will be when \(r = 0\) (at \(\theta = 0\) or \(\pi\)). Again, sketch this graph on the same plane.

Step 4: Graph for k = 3

The equation changes to \(r=2+3\sin\theta\). Here, the sin function contributes even more to the radius. This results in an even wider loop. The highest point on the loop will be when \(r = 5\) (at \(\theta = \pi/2\) or \(3\pi/2\)), and the lowest point will be when \(r = -1\) (at \(\theta = 0\) or \(\pi\)). Sketch this graph on the same plane.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to the traditional Cartesian coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin in Cartesian coordinates) is called the pole, and the ray from the pole in the reference direction is the polar axis.

In polar coordinates, a point is described by a pair \(r, \theta\), where \(r\) represents the radius or the distance from the pole, and \(\theta\) represents the angle in radians from the polar axis, measured in the counter-clockwise direction.

To support students' understanding of polar equations during homework exercises, it may be useful to employ interactive tools such as graphing calculators or software that can visually represent changes in the equations as values adjust. This visual representation aids in grasping the relationship between radius, angle, and the overall shape of the graph.

Sine Function

The sine function is a fundamental trigonometric function that emerges in a variety of mathematical contexts, including polar equations. It is a periodic function, meaning it repeats its values at regular intervals, specifically every \(2\pi\) radians.

For a given angle \(\theta\), the sine function, denoted as \(\sin(\theta)\), quantifies the y-coordinate of the point on the unit circle at that angle from the positive x-axis.

In the context of polar coordinates, the sine function can modulate the radius for various values of \(\theta\). In the classroom, understanding the sine function's wave-like pattern and its impact on polar graphs can be improved through activities such as plotting points of a sine wave on Cartesian coordinates before linking these observations back to polar forms. This enhances comprehension of how the sine function influences the shape and size of polar graphs.

Radius Modulation

Radius modulation in polar graphs refers to the variation of the radius based on a trigonometric function, like the sine function. In the given exercise \(r = 2 + k\sin\theta\), the coefficient \(k\) before the sine function modulates how much the sine wave affects the radius.

When \(k=0\), the radius remains constant, and you get a circle. As \(k\) increases, the impact of the sine function becomes more pronounced, resulting in various sizes and shapes of loops or petals around the graph's polar axis. These can be described as limaçon graphs. This modulation means that for different angles \(\theta\), the distance \(r\) from the pole fluctuates producing a richer variety in polar graphs.

Encouraging students to increment \(k\) slowly while observing the changes in the graph can help them better understand the relationship between the sine function's amplitude and the modulation of the radius, making the behavior of these equations more tangible.