Problem 61 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=5$$

Step-by-Step Solution

Answer

The graph is a circle with radius 5 centered at the origin.

Step 1: Identify the Polar Equation

We are given the polar equation \( r = 5 \). This means that for every angle \( \theta \), the radius \( r \) is always 5.

Step 2: Understand the Equation

The equation \( r = 5 \) represents a circle in the polar coordinate system. Since the radius is constant (5), the circle is centered at the origin (0,0) with a radius of 5.

Step 3: Sketch the Circle

To sketch the graph, draw a circle with center at the origin (0,0) and a radius of 5 units in all directions in the polar coordinate plane. This will look like a conventional circle drawn on Cartesian coordinates, but centered at the polar origin.

Key Concepts

Polar Equation

Understanding a polar equation is key to drawing graphs in polar coordinates. Unlike the Cartesian system where we use coordinates \(x,y\), the polar system uses a radius \(r\) and angle \(\theta\). In our exercise, we examine \(r = 5\). This is a simple polar equation, yet different from Cartesian equations involving \(x\) and \(y\). Here, \(r\) represents the distance from the origin, which is \(0,0\) in polar coordinates.
Polar equations come in various forms but typically reference the relationship between \(r\) and \(\theta\). When \(r\) is a constant, like in \(r = 5\), it indicates a fixed distance, helping us identify what kind of shape to expect, which can often lead to a straightforward graphing process.
By using polar equations, we can easily represent shapes centered around a point, making this system increasingly useful for circular and rotational symmetry.

Graph Sketching

Sketching graphs from polar equations involves different, yet rewarding skills. Let's break it down. When tasked with graphing \(r = 5\), recognize that each \(\theta\) results in the same \(r\). This repetition across all angles traces out a simple circle.
Steps to sketch:

• Recognize \(r\) as constant. It defines a fixed radius for your shape.
• Visualize that \(\theta\) sweeps around a full circle from 0 to 2\(\pi\), keeping the radius unchanged, naturally forming a circle.
• Draw a circle that extends outward equally from the center.

Approaching graph sketching with these steps allows for better spatial understanding and connects closely to real-world phenomena like radar, which employs polar coordinates quite effectively.

Circle in Polar Coordinates

Circles often take a clean, intuitive form in polar coordinates. When given an equation like \(r = 5\), we deal with a circle whose radius is consistently 5, centered at the origin. This equidistant property from the center at all angles stems from the constant \(r\).
Polar circles can be applied to areas such as physics, engineering, and navigation. Here's why:

• In navigation, they help depict areas of influence or signals from a central point.
• In physics, phenomena exhibiting rotational symmetry can be easily analyzed using polar coordinates.
• Engineering applications include material stress analysis where circular patterns are involved.

From this, you see how a circle in polar coordinates is more than just a graph—it's a versatile tool.