Problem 83 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=2 $$

Step-by-Step Solution

Answer

This equation represents a circle with radius 2.

Step 1: Recognize the given equation

The given equation is in rectangular coordinates. It is given by \( r = 2 \).

Step 2: Identify the relationship between polar and rectangular coordinates

In rectangular coordinates, any point can be represented as \((x, y)\). In polar coordinates, a point is represented as \((r, \theta)\).

Step 3: Interpret the polar coordinate equation

The equation \( r = 2 \) in polar coordinates means that the distance from a point to the origin is always 2, forming a circle of radius 2.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are the most commonly used coordinate system for graphing points on a plane. In this system, any point is represented by an ordered pair \( (x, y) \). The \(x\)-coordinate represents the horizontal distance from the origin, while the \(y\)-coordinate represents the vertical distance. Think of graph paper where you plot points: the place you put a point is decided by these \(x\) and \(y\) values.

Rectangular coordinates make it straightforward to perform algebraic operations but can sometimes be less intuitive when working with circles or angles.

Coordinate Transformation

Coordinate transformation is the process of converting coordinates from one system to another. In this case, we are converting rectangular coordinates to polar coordinates.

Pole polar coordinates represent a point using \( r \) (distance from the origin) and \( \theta \) (angle from the positive x-axis). To convert between these two systems, we use the following formulas:

• From rectangular to polar: \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}( \frac{y}{x} ) \).
• From polar to rectangular: \( x = r \cos( \theta ) \) and \( y = r \sin( \theta ) \).

These conversions are crucial in solving problems that involve both coordinate systems or need simpler forms for specific shapes or functions.

Representation of Points

The way we represent points will change depending on the coordinate system we use.

Let's look at the given exercise: the equation \( r = 2 \) in polar coordinates. This represents all points that are a distance of 2 from the origin. If you plot this on the plane, it forms a circle with a radius of 2 centered at the origin.

In the rectangular coordinate system, such an equation doesn't appear as simple; here, it would be expressed as: \[ x^2 + y^2 = 2^2 \] which is \( x^2 + y^2 = 4 \).

Remember, the choice of coordinate system can significantly simplify our equations and our understanding of shapes.