Problem 45 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=5 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Polar Equation

The given polar equation is \( r = 5 \). In polar coordinates, any point on the graph is determined by \( r \) (the distance from the origin) and \( \theta \) (the angle with the positive x-axis).

Step 2: Identify Graph Type

Since the equation is \( r = 5 \), which indicates a constant distance from the origin, this represents a circle centered at the origin with a radius of 5.

Step 3: Graph the Circle

To sketch the graph of the equation \( r = 5 \), draw a circle centered at the origin of the polar coordinate system with a radius of 5. All points on this circle are exactly 5 units away from the origin, irrespective of the angle \( \theta \).

Step 4: Label the Graph

Label the circle properly on the polar coordinate system with a radius of 5. Indicate that each point on the circle satisfies \( r = 5 \).

Key Concepts

Graphing Polar Equations

Graphing polar equations might initially seem daunting, but it's actually quite straightforward once you grasp the basics. In the polar coordinate system, each point on a plane is determined by a distance from a reference point known as the origin and an angle from a reference direction, usually the positive x-axis.

• The \( r \) value represents the distance from the origin, much like the length of a radius in a circle.
• The \( \theta \) value represents the angle, providing directionality to the point in relation to the origin.

When graphing a polar equation like \( r=5 \), you aren't flustered with too many variables as you might be in Cartesian coordinates. Here, the task is simpler—just focus on the radial distance given by \( r \). The angle \( \theta \) is not specified, which means that this fixed distance of 5 forms an infinite number of positions around the origin, creating a circle.
In this way, polar equations become a fascinating intersection of geometry and algebra, offering a unique way to express familiar shapes like circles, spirals, and roses.

Distance from Origin

The polar coordinate system fundamentally revolves around the concept of distance from the origin. Here, the distance \( r \) defines how far a point is from the origin, which is somewhat akin to the hypotenuse in a right-angle triangle.
This distance is straightforward to determine when given as a constant in a polar equation, such as \( r=5 \). Unlike Cartesian coordinates, which use both \( x \) and \( y \) axes to determine position, polar coordinates focus primarily on \( r \), simplifying the distance concept to a single variable.

• For \( r=5 \), every point on the graph lies exactly 5 units away from the origin, forming a perfect circle.
• There is no angle mentioned because \( r \) alone describes the entire shape.

By considering distance in this way, we can easily draw shapes like circles without switching back and forth between \( x \) and \( y \) values, simplifying many graphing situations encountered in geometry.

Circle in Polar Coordinates

The simplicity of a circle in polar coordinates can be truly captivating. A polar equation like \( r=5 \) is inherently straightforward as it describes a circle centered at the origin with a fixed radius—in this case, the radius is 5.

• The term ‘radius’ here refers to \( r \), the constant distance from the origin.
• As \( \theta \) varies, every point around this fixed distance draws the boundary of a circle.

Thus, the equation \( r=5 \) encompasses all those points in a plane that maintain that consistent distance of 5 units from the origin.
This visualization contrasts with the more complex circle equation in Cartesian coordinates:

• In Cartesian, \( x^2 + y^2 = 25 \) represents the same circle, derived from a Pythagorean interpretation.

Utilizing polar equations not only simplifies the process of graphing but also emphasizes the intrinsic relationship between geometry and algebra, showcasing the circle in its purest form.