Question

Describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$ r=3 $$

Short answer

The graph is a circle centered at the origin with a radius of 3. The rectangular equation is \(x^2 + y^2 = 9\).

Step-by-step solution

Understanding the Polar Equation

The given polar equation is \( r = 3 \). In polar coordinates, \( r \) represents the distance from the origin to any point on the graph. This equation means that the distance \( r \) is constant at 3 for all angles \( \theta \).

Describe the Graph

Since \( r = 3 \) for all values of \( \theta \), the graph represents a circle centered at the origin with a radius of 3. This is because every point that is 3 units away from the origin will lie on a circle.

Convert to Rectangular Coordinates

In polar coordinates, the conversion to rectangular coordinates follows the formulas: \[ x = r \cos(\theta) \] and \[ y = r \sin(\theta) \]. Substituting \( r = 3 \) into these equations gives: \( x = 3 \cos(\theta) \) and \( y = 3 \sin(\theta) \).

Derive the Rectangular Equation

To eliminate \( \theta \), square both the \( x \) and \( y \) equations: \( x^2 = 9 \cos^2(\theta) \) and \( y^2 = 9 \sin^2(\theta) \). Add these two equations: \[ x^2 + y^2 = 9(\cos^2(\theta) + \sin^2(\theta)) = 9. \] This is because \( \cos^2(\theta) + \sin^2(\theta) = 1 \). Thus, the rectangular form of the equation is \( x^2 + y^2 = 9 \), which is the equation of a circle centered at the origin with a radius 3.

Key concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a system that uses two perpendicular axes to define the position of points in the plane. Each point is specified by an ordered pair \(x, y\), where 'x' and 'y' represent the horizontal and vertical distances of the point from the origin (0, 0) respectively.

• **X-axis:** The horizontal axis.
• **Y-axis:** The vertical axis.

Polar coordinates, on the other hand, use a different system based on a radius 'r' and an angle \(\theta\) from the positive x-axis. Converting between polar and rectangular coordinates involves understanding their relationship:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

These equations help us translate a position in polar coordinates into the rectangular system, allowing us to graph points in a familiar plane format.

Equation Conversion

Equation conversion is the process of changing equations from one coordinate system, such as polar coordinates, to another system like rectangular coordinates.
This process can make it easier to analyze and visualize equations because many graphs are more intuitive in the rectangular plane.For example, consider the polar equation \( r = 3 \). To convert this polar equation to its rectangular form, we use the known relationships between polar and rectangular coordinates. Plugging \( r = 3 \) into:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

we get:

• \(x = 3 \cos(\theta)\)
• \(y = 3 \sin(\theta)\)

By squaring these equations and summing them up, we find that \(x^2 + y^2 = 9\), which represents a circle with radius 3 centered at the origin. This conversion shows how seemingly complex equations in polar form can transform into simpler shapes in rectangular form.

Graph Description

Describing the graph of an equation involves determining the shape and position of the curve represented by the equation.
For the polar equation \( r = 3 \), we describe the graph by understanding that \( r \), the distance from the origin, is constant for all angles \( \theta \). This consistency suggests a set of points all equidistant from a central point, forming a perfect circle.To provide a detailed description:

• **Shape:** The graph is a circle.
• **Center:** The circle is centered at the origin (0, 0).
• **Radius:** The radius of the circle is 3.
• **Symmetry:** Being a circle, it is symmetric about any line through its center.

In rectangular coordinates, this description translates to the equation \(x^2 + y^2 = 9\). Here, recognizing this pattern reinforces the understanding of how polar coordinates convey geometric shapes differently from rectangular coordinates.