Problem 21 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=6 \sin \theta $$

Step-by-Step Solution

Answer

The graph of \( r = 6\sin\theta \) is a circle centered at (0, 3) with radius 3. The rectangular form is \( x^2 + (y - 3)^2 = 9 \).

Step 1: Transform the Polar Equation to Rectangular Coordinates

The given polar equation is \( r = 6 \sin \theta \). We will use the conversion formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). From \( y = r \sin \theta \), we can replace \( r \sin \theta = y \), so \( r = \frac{y}{\sin \theta} \). Therefore, the equation becomes \( r = 6 \cdot \frac{y}{r} \), resulting in \( r^2 = 6y \). But since \( r^2 = x^2 + y^2 \), then \( x^2 + y^2 = 6y \). The rectangular equation is \( x^2 + (y - 3)^2 = 9 \).

Step 2: Identify the Shape in Rectangular Coordinates

In rectangular coordinates, the equation \( x^2 + (y - 3)^2 = 9 \) represents a circle. The circle has a center at \((0, 3)\) and a radius of \(3\). This is consistent with the form \((x - h)^2 + (y - k)^2 = r^2\) for a circle.

Step 3: Sketch the Polar Graph

For the polar equation \( r = 6 \sin \theta \), the circle is traced by varying \( \theta \). Since \( \sin \theta \) varies between -1 and 1, \( r \) will range from 0 to 6, describing a circle centered at \((0, 3)\) with a maximum distance from origin (radius) of 3 in the positive y-direction. On polar coordinates, plot points at several angles and join smoothly to form this circle.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use a grid system where each point is defined by an x-coordinate (horizontal position) and a y-coordinate (vertical position). This system is familiar because it matches the way we draw on graph paper. For example, the point (3, 4) lies 3 units along the x-axis and 4 units up the y-axis.

Understanding rectangular coordinates helps us visualize equations in terms of geometric shapes, like lines, circles, and parabolas. Equations like \(x^2 + (y - 3)^2 = 9\) describe circles precisely by their center and radius on this grid.

• **Center of the Circle:** Given by the point \((h, k)\); here it's \((0, 3)\).
• **Radius of the Circle:** Derived from the equation \(r^2\); in this case \(r = 3\).
Recognizing these elements helps students connect algebraic expressions to geometric figures.

Polar to Rectangular Conversion

Converting between polar and rectangular coordinates is essential when graphing or describing shapes defined in different systems. In polar coordinates, each point is given by \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis.

To convert a polar equation like \(r = 6 \sin \theta\) to rectangular form, we rely on the conversion formulas:

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

In the solution, we transform \(r = 6 \sin \theta\) into \(x^2 + y^2 = 6y\) to help plot the graph within the Cartesian system. Manipulating these formulas bridges the gap between the two systems.

Graphing Polar Equations

Graphing polar equations involves plotting points in a circular layout using polar coordinates \((r, \theta)\). Unlike rectangular coordinates that plot on a grid, polar plots are based on circles around a central point.

For the equation \(r = 6 \sin \theta\), we observe that as \(\theta\) changes, \(r\) defines the radius of the circle traced out by the movement. Since the equation involves \(\sin \theta\), it describes a full circle centered at \((0, 3)\) with a radius of 3. Here's the process:

• Calculate \(r\) for various \(\theta\) angles.\(\theta\) typically ranges from \(0\) to \(2\pi\).
• Plot each point by calculating its distance from the origin and the angle.
• Connect these points to form the circle.

This method visualizes how polar coordinates can beautifully and easily represent curves and shapes, offering a complementary perspective to rectangular graphs.