Problem 43 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r-6 \sin \theta=0$$

Step-by-Step Solution

Answer

The graph is a circle centered at (0, 3) with radius 3.

Step 1: Understand the Polar Equation

The given polar equation is \( r - 6 \sin \theta = 0 \). This can be simplified to \( r = 6 \sin \theta \). It represents a circle in the polar coordinate system.

Step 2: Convert Polar Equation to Cartesian Equation

To convert the polar equation to Cartesian coordinates (\(x, y\)), we use the relationships \( r = \sqrt{x^2 + y^2} \), \( x = r \cos \theta \), and \( y = r \sin \theta \). From \( r = 6 \sin \theta \), multiplying both sides by \(r\), we get \( r^2 = 6r \sin \theta \), which substitutes to \( x^2 + y^2 = 6y \).

Step 3: Rearrange Cartesian Equation

Rearrange the equation \( x^2 + y^2 = 6y \) to solve for a circle. Subtract \(6y\) from both sides to obtain \( x^2 + y^2 - 6y = 0 \).

Step 4: Complete the Square

Complete the square to convert \( x^2 + y^2 - 6y = 0 \) into the standard form of a circle. Rewrite the equation as \(x^2 + (y^2 - 6y) = 0\). Complete the square for \(y\): \( y^2 - 6y = (y - 3)^2 - 9\). Thus, \( x^2 + (y - 3)^2 = 9 \), which is the equation of a circle centered at \((0, 3)\) with radius \(3\).

Step 5: Sketch the Graph in the Polar Plane

The polar equation \( r = 6 \sin \theta \) translates to a circle in the Cartesian system with its center at \((0, 3)\) and radius \(3\). In polar coordinates, it is centered above the polar axis at \( r = 3 \), and the circle extends around this central point.

Key Concepts

Polar Coordinates

Polar coordinates provide a method to locate points in a plane using a distance and an angle. Instead of the familiar Cartesian coordinate system, which uses

• the horizontal distance (\(x\)) and
• the vertical distance (\(y\))

from a point called the origin, polar coordinates describe a point by its distance from a fixed point (the pole, usually the origin) and the angle from a fixed direction (usually the positive x-axis).

In polar coordinates, a point is identified as \((r, \theta)\) where:

• \(r\) represents the radial distance from the pole, and
• \(\theta\) is the angle measured in radians from the positive x-axis.

The position of a point can be described with infinite possibilities of \(\theta\) since one full rotation around the pole adds\(2\pi\)to \(\theta\).
Polar coordinates are especially useful in defining curves and shapes like circles and spirals easily, providing simplicity in certain mathematical contexts where Cartesian coordinates get complex.
For instance, the equation \(r = 6 \sin \theta\) describes a circle in polar coordinates.

Cartesian Coordinates

The Cartesian coordinate system is a method to graphically represent points within a plane using a pair of numbers that define their horizontal and vertical position relative to two perpendicular axes.
In this system, any point can be expressed as \((x, y)\), where:

• \(x\) is the horizontal coordinate measuring the displacement from the y-axis, and
• \(y\) is the vertical coordinate measuring the displacement from the x-axis.

Converting from polar coordinates (\((r, \theta)\)) to Cartesian coordinates (\((x, y)\)) involves the equations:

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

These transformations allow for mathematical equations in polar form to be expressed in a familiar \((x, y)\) space, which is useful for visualization or solving problems that rely on perpendicular directions. For example, a circle can be effectively graphed in both systems, with its properties more recognizable in Cartesian coordinates after conversion.

Equation of a Circle

The equation of a circle in Cartesian coordinates is derived to express the consistent distance from the center to any point on the circle. It is typically presented in its standard form:\[(x - h)^2 + (y - k)^2 = r^2\]where:

• \((h, k)\) represents the center of the circle, and
• \(r\) represents the radius.

From our previous conversion, the circle's equation was rearranged to \(x^2 + (y - 3)^2 = 9\), indicating:

• The circle is centered at \((0, 3)\)
• with a radius of \(3\)

This conversion allows us to see the shape and size of the circle more clearly as it stands out visibly in the \((x, y)\) plane.
Circles in Cartesian form are essential in geometry, physics, and engineering because they offer straightforward calculations involving tangents, areas, and circumferences.