Problem 33 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=1+\cos \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r=1+\cos \theta\) is a circle with its center at (1, 0) and a radius of 1.

Step 1: Understanding Polar Coordinates

Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point (r) and an angle from a reference direction (\(\theta\)). The angle \(\theta\) is commonly measured in radians and the radius \(r\) is usually considered to be non-negative.

Step 2: Understanding the Given Polar Equation

The given polar equation is \(r=1+\cos \theta\). This format of the equation shows that for each angle \(\theta\), the radius \(r\) will be \(1 + \cos \theta\). Depending on the value of \(\cos \theta\), the radius will change.

Step 3: Sketching the graph

To draw the graph of a polar equation, the easiest way often involves plotting some key points and then sketching the curve that joins them. Begin by considering various angles, such as \(\theta = 0, \pi/2, \pi, 3\pi/2\), and \(2\pi\), and calculating the corresponding values of \(r\). After plotting these points, draw a smooth curve through them to represent the graph.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that define relationships between the radius and angle in polar coordinates. These equations are crucial in describing curves that may look complex when portrayed in a Cartesian coordinate system, yet appear simple in polar form. For example, a circle centered at the origin with a radius of 2 can be described by the polar equation \( r = 2 \). Here, \( r \) represents the distance from the origin, which remains constant for all angles \( \theta \).

In polar equations like \( r = 1 + \cos \theta \), the radius \( r \) varies as a function of the angle \( \theta \). This means the distance from any point on the curve to the origin isn't constant but changes depending on the angle. Polar equations are invaluable for capturing the symmetry and periodicity of curves, especially those not easily expressed in rectangular form.

Graphing Polar Coordinates

Graphing polar coordinates involves plotting points based on their radial distance \( r \) from the origin and their angle \( \theta \), measured from the positive x-axis. This differs significantly from plotting points on a Cartesian plane, where each point is dictated by an \(x\)- and \(y\)-coordinate.

To graph polar equations like \( r = 1 + \cos \theta \), it is helpful to calculate values of \( r \) for several key angles, such as \( \theta = 0, \pi/2, \pi, 3\pi/2, \) and \( 2\pi \). Each of these calculations will give a distance \( r \) that, combined with \( \theta \), locates a point on the polar grid.

• At \( \theta = 0 \), \( r = 1 + \cos(0) = 2 \).
• At \( \theta = \pi/2 \), \( r = 1 + \cos(\pi/2) = 1 \).
• At \( \theta = \pi \), \( r = 1 + \cos(\pi) = 0 \).
• At \( \theta = 3\pi/2 \), \( r = 1 + \cos(3\pi/2) = 1 \).
• At \( \theta = 2\pi \), \( r = 1 + \cos(2\pi) = 2 \).

Connecting these points smoothly gives the complete graph. This particular equation forms a "limacon" shape, which is characteristic of polar graphs of this form.

Cosine Function in Polar Form

The cosine function plays a pivotal role in polar graphs, especially when expressed as part of the polar equation \( r = 1 + \cos \theta \). The function modifies the distance \( r \) from the origin based on the angle \( \theta \). Understanding this alteration by the cosine function is key to predicting and sketching the curve accurately.

The values of the \( \cos \theta \) function vary between -1 and 1, and thus, directly influence the radius \( r \) in the equation. This variation causes the curve to expand and contract as \( \theta \) changes. In our example, when \( \theta = \pi \), the value of \( \cos(\pi) \) becomes -1, making \( r = 0 \). This brings the point right back to the origin, leading to an indentation or cusp in the graph, typical of a limacon with an inner loop.

Notice that cosine's periodicity implies that certain shapes, such as limacons, exhibit symmetry, often about the polar axis or another line. Recognizing these symmetrical properties helps when sketching and understanding these distinct polar curves.