Problem 86 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\frac{3}{3-\cos \theta} $$

Step-by-Step Solution

Answer

The polar equation is already simplified as \(r=\frac{3}{3-\frac{\theta}}\).

Step 1: Understand Polar Coordinates

Recognize that polar coordinates \(r, \theta\) include the radius \(r\) and the angle \(\theta\) from the positive x-axis.

Step 2: Identify Given Polar Equation

Given polar equation is \(r=\frac{3}{3-\frac{\theta}}\).

Step 3: Simplify the Polar Equation

Check if the polar equation is already simplified. In this case, it is, so no further simplification is necessary.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to map points on a plane using two perpendicular axes. These axes are typically labeled as the x-axis (horizontal) and the y-axis (vertical). Each point is described by an ordered pair \(x, y\).
The x-coordinate represents the point's horizontal position, while the y-coordinate represents its vertical position.
This system is incredibly useful for plotting points, drawing curves, and performing many kinds of calculations in algebra and geometry.
For example, the point \(-3, 2\)\ means that you move 3 units left on the x-axis and 2 units up on the y-axis.

Conversion to Polar Form

Polar coordinates offer a different approach to locating points using a radius and an angle. In polar coordinates, each point is represented by \(r, \theta \), where:

• \(r\) is the distance from the origin to the point.
• \(\theta\) is the angle formed with the positive x-axis.

To convert rectangular coordinates \(x, y\) to polar coordinates \(r, \theta \):
Use the formulas:

• \ r = \sqrt{x^2 + y^2}\
• \ \theta = \arctan\left( \frac{y}{x} \right) \

Example: For point \(3, 4\):

• Radius: \ r = \sqrt{3^2 + 4^2} = 5 \
• Angle: \ \theta = \arctan\left( \frac{4}{3} \right) \

Therefore, the polar coordinates are \, \left( 5, \arctan \left( \frac{4}{3} \right) \right) \.

Simplification of Polar Equations

Simplifying polar equations often involves making them easier to understand or solve. Sometimes the given equations are already simplified, while other times they may need more work.
For instance, let's examine the given equation \: \ r = \frac{3}{3 - \cos\theta} \
The equation specifies the radius \(r\) in terms of the angle \(\theta\).
Steps for simplification:

• Check whether the terms on the right-hand side can be simplified further. In this case, they cannot.
• Ensure that there are no unnecessary terms or factors.
• Rewrite the equation in a more straightforward form if possible.

Since the original equation given is already in its simplest form, no further action is needed.
Understanding and checking for potential simplifications help reduce computational complexity and make equations easier to work with.