Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=3 \cos \theta $$

Step-by-Step Solution

Answer

The graph is a circle centered at (1.5, 0) with radius 1.5.

Step 1: Understanding the Polar Equation

The given polar equation is \( r = 3 \cos \theta \). This means that the distance from the pole (origin) changes as the angle \( \theta \) changes. To plot a few key points, we can evaluate the equation at specific angles to better understand the shape.

Step 2: Evaluate the Polar Equation at Key Angles

To get some key points, substitute different values for \( \theta \). For example, when \( \theta = 0 \), \( r = 3 \cdot \cos(0) = 3 \). When \( \theta = \frac{\pi}{2} \), \( r = 3 \cdot \cos\left(\frac{\pi}{2}\right) = 0 \). And when \( \theta = \pi \), \( r = 3 \cdot \cos(\pi) = -3 \). This means that at \( \pi \), the point is still on the positive x-axis but has wrapped around.

Step 3: Identify the Shape of the Graph

The polar equation \( r = 3 \cos \theta \) is in the form of \( r = a \cos \theta \) which describes a circle centered on the x-axis. The circle's center is at \( (a/2, 0) = (3/2, 0) \), and the radius is \( a/2 = 3/2 \). Therefore, this equation describes a circle with radius 1.5 centered at (1.5, 0).

Step 4: Sketch the Graph

Now sketch the circle based on the information from previous steps: it's centered at \( (1.5, 0) \) on the polar plane (with the pole at the origin) and has a radius of 1.5. The circle will intersect the pole when \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).

Key Concepts

Polar Equations

Polar equations are mathematical expressions defining the relationship between the radius, \(r\), from a fixed point called the origin, or pole, and the angle, \(\theta\), measured from the polar axis. These equations are particularly useful for graphing curves that are symmetric or circular in nature, helping to represent curves more directly than Cartesian equations might.

In the given exercise, the polar equation \( r = 3 \cos \theta \) depicts how the radius varies with different angles \(\theta\). When using polar equations, it's common practice to evaluate the expression at various angles to identify key points on the graph. This provides insights into the shape or pattern that the polar curve assumes. By testing angles like \(0, \frac{\pi}{2}, \pi\), and \(\frac{3\pi}{2}\), you can visualize how the equation forms a specific pattern, like a circle in this instance.

Polar equations are instrumental for tasks such as:

• Modeling phenomena with circular or spiral patterns
• Designing complex curves in fields such as physics and engineering
• Evaluating systems that rely on rotation and circular symmetry

Circle Graphing

Graphing a circle in polar coordinates can initially seem daunting but is simplified greatly by understanding polar equations. For the polar equation \( r = 3 \cos \theta \), it forms a circle because it fits the general form \( r = a \cos \theta \), where \(a\) is a constant.

The center of this circle lies on the x-axis (or polar axis in polar terms) at \((\frac{a}{2}, 0)\), and it has a radius of \(\frac{a}{2}\). Therefore, for our equation, the center is at \((1.5, 0)\) with a radius 1.5 units. This indicates that as \(\theta\) changes, the value of \(r\) traces out a circle in the polar plane.

Key elements to understand when graphing polar circle equations include:

• Center and Radius: Determine these from the coefficient of the trigonometric function.
• Intersections: Points where the circle intersects the polar axis, typically at right angles such as \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).
• Symmetry: Reflection across the polar axis is common with these curves.

Trigonometric Functions

The role of trigonometric functions in polar equations is central as they describe the relationship between the radius and angle. In our exercise, the trigonometric function \(\cos \theta\) is used within the equation \(r = 3 \cos \theta\), highlighting its use for determining both the circle's radius and its shape.

Trigonometric functions like sine and cosine can express not just circular movements, but also aid in understanding oscillations, cycles, and angular relationships.

Key aspects of trigonometric functions in polar contexts:

• Periodicity: Functions such as \(\cos \theta\) repeat every \(2\pi\), which helps predict the polar curves' pattern.
• Amplitude: The coefficient of \(\cos \theta\) (like the 3 in our example) influences the size or range of movement, determining the radius in circle equations.
• Angle Dependency: The value of \(r\) changes as \(\theta\) changes, showcasing how trigonometric functions specify polar coordinates.

Understanding these principles assists greatly in predicting and sketching accurate graphs of polar equations, providing clarity on how trigonometric properties influence graphical shapes.