Problem 109 - Precalculus with Limits

Question

In Exercises 109-118, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. \(r=6\)

Step-by-Step Solution

Answer

The rectangular equation for the given polar equation \(r=6\) is \(x^2 + y^2 = 36\), and it represents a circle centered at the origin with a radius of 6 units.

Step 1: Analyze the given polar equation

The polar equation is \(r = 6\). This represents a circle with radius 6. In polar coordinates, when r is constant, it's always a circle centered at the origin.

Step 2: Convert the polar equation to a rectangular equation

We can convert the polar equation to a rectangular equation using the equation relating the polar and rectangular systems, \(r^2 = x^2 + y^2\). For \(r = 6\), squaring both sides we get \(r^2 = 36\), so in rectangular coordinates, this is \(x^2 + y^2 = 36\)

Step 3: Sketch the graph of the rectangular equation

The equation \(x^2 + y^2 = 36\) represents a circle with a radius of 6 units centered at the origin (0,0). Sketching this will result in a circle centered at the origin, with points on the circle 6 units away from the center in all directions.

Key Concepts

Rectangular Equations

Rectangular equations are a fundamental concept in mathematics, often expressed in the Cartesian coordinate system using

• x and y values.
• These coordinate systems help us visualize and solve problems through graphs and geometric shapes.

In the exercise, the polar equation given is \( r = 6 \). Polar equations describe systems based on a radius and angle.
To convert it into a rectangular equation, we use the formula relating polar and rectangular coordinates:
\( r^2 = x^2 + y^2 \). Since \( r = 6 \), substituting \( r \) gives us \( r^2 = 36 \).
Thus, the equation becomes \( x^2 + y^2 = 36 \). This is a standard equation for a circle with a radius of 6 and centered at the origin in the xy-plane, representing a geometric figure that can easily be graphed.

Coordinate Conversion

Coordinate conversion is essential when working between different systems like polar and rectangular.

• Polar coordinates use radius (r) and angle (θ) measurements.
• Rectangular coordinates use the Cartesian (x, y) system.

Understanding how to switch between these is vital for solving equations or graphing.
In polar coordinates, the position of a point is determined relative to a central point (origin) and an angle.
One straightforward formula for converting from polar to rectangular is \( x = r \cdot \cos(θ) \) and \( y = r \cdot \sin(θ) \).
For equations where \( r \) is constant, like \( r = 6 \), we can simplify using \( x^2 + y^2 = r^2 \), yielding: \( x^2 + y^2 = 36 \). This conversion is useful when interpreting data or drawing graphs in both coordinate systems, allowing for better versatility and understanding of geometric relationships.

Graph Sketching

Graph sketching involves drawing a visual representation of mathematical equations on a coordinate plane. It helps us understand and interpret equations by visualizing the shapes and patterns they create. This skill is particularly beneficial in interpreting rectangular equations.

• To sketch the circle from our exercise, you start with the center at the origin (0,0).
• The radius is 6, which means every point on the circle is exactly 6 units from the center.

To sketch, imagine extending a measuring tool in all directions from the center up to 6 units, forming a complete circle.
This process is aided by understanding that the circle defined by \( x^2 + y^2 = 36 \) will perfectly encompass all points that match this distance.
Visualizing these graphs reinforces mathematical intuition and aids in solving more complex problems by decoding relationships within equations.