Question

Find the Cartesian equations of the graphs of the given polar equations. $$ r=3 $$

Short answer

The Cartesian equation is \( x^2 + y^2 = 9 \).

Step-by-step solution

Understanding Polar Equations

The polar equation given is \( r = 3 \). This indicates that the distance from the origin (pole) to any point on the graph is always 3 units. In polar coordinates, this usually represents a circle centered at the origin with a radius of 3.

Converting Polar to Cartesian Coordinates

To convert the polar equation \( r = 3 \) into a Cartesian equation, we use the relationship between polar and Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Substituting Values into Cartesian Formulas

By substituting \( r = 3 \) into the equations, we get: \( x = 3 \cos \theta \) and \( y = 3 \sin \theta \).

Expressing in Terms of Circle Equation

The above values of \( x \) and \( y \) for all \( \theta \) represent points on a circle. We recall that in Cartesian coordinates, the equation of a circle centered at the origin with radius \( r \) is \( x^2 + y^2 = r^2 \).

Formulating the Final Cartesian Equation

Since \( r = 3 \), we substitute into the circle equation format: \( x^2 + y^2 = 3^2 \), which simplifies to \( x^2 + y^2 = 9 \). This is the Cartesian equation of the circle.

Key concepts

Polar Coordinates

Polar coordinates are a way of expressing the location of a point in the plane using a distance and an angle. Imagine you are standing at the origin and looking out in any direction. The polar coordinate system measures how far you have to walk (denoted as \( r \)), and at what angle relative to the positive x-axis (denoted as \( \theta \)), you must turn to reach your point.
With polar coordinates, the concept becomes intuitive when picturing points in the context of navigation, such as a ship or a plane, where direction and distance are crucial. The polar equation \( r = 3 \) suggests that every point on this graph maintains a consistent distance of 3 units from the origin, forming a perfect circle. This is a hallmark of how circles are represented in polar coordinates.

Cartesian Coordinates

In contrast to polar coordinates, Cartesian coordinates describe a point by its horizontal and vertical distances from a reference point, typically the origin. Each point is specified as \( (x, y) \), representing how far you need to go along the x-axis and y-axis to reach your point.
Cartesian coordinates are widely used due to their straightforward nature, especially in applications like graphing data and geometry. The transition from polar to Cartesian provides a visual grid-like view of the geometry behind shapes and equations. For our circle, identifying its equation in Cartesian form will show how it fits within the classic x-y plane being a circle of radius 3 centered at the origin.

Circle Equation

The equation of a circle in Cartesian coordinates is conveniently written as \( x^2 + y^2 = r^2 \). Here, \( r \) is the radius of the circle, and if the circle is centered at the origin, this formula perfectly describes it.
When we have the polar equation \( r = 3 \), converting it into a Cartesian equation involves recognizing that the circle is centered at the origin, with a radius of 3. Thus the equation \( x^2 + y^2 = 9 \) is straightforward. This tells us that every point \((x,y)\) on the circle maintains the equation, demonstrating the power of defining geometry using algebra.

Coordinate Conversion

Converting between polar and Cartesian coordinates can seem magical, but it's a simple transformation based on trigonometric relationships. The formulas \( x = r \cos \theta \) and \( y = r \sin \theta \) bridge the two systems.
For the circle defined by \( r = 3 \), you substitute into these equations to solve for \( x \) and \( y \). As \( \theta \) varies, \( x \) and \( y \) trace out all the points on the circle as described previously. This conversion not only helps in visualization but also in understanding how shapes like circles can be expressed within different coordinate systems.

• Formula for \( x \): \( x = 3 \cos \theta \)
• Formula for \( y \): \( y = 3 \sin \theta \)
• Resulting Cartesian equation: \( x^2 + y^2 = 9 \)

Coordinate conversion is a fundamental skill in mathematics and engineering, useful in translating complex ideas into comprehensible forms.