Problem 55 - Precalculus : Building Concepts and Connections

Question

Convert each of the given polar equations to rectangular form. $$r=3$$

Step-by-Step Solution

Answer

The rectangular form of the given polar equation \( r = 3 \) is \( x^2 + y^2 = 9 \)

Step 1: Identify the given polar coordinates

In the given equation \( r = 3 \), the polar coordinate \( r \) is provided and it is 3.

Step 2: Apply the Cartesian conversion formula

To convert this polar equation into a rectangular equation, we use the conversion principle of \( r = \sqrt{x^2 + y^2} \) to replace \( r \) by the expression \( \sqrt{x^2 + y^2} \).

Step 3: Solve the equation

We replace \( r \) in the given polar equation \( r = 3 \) with \( \sqrt{x^2 + y^2} \), getting \( \sqrt{x^2 + y^2} = 3 \). Squaring both sides of the equation, we finally obtain \( x^2 + y^2 = 9 \), which is the rectangular form of the given polar equation

Key Concepts

Polar Coordinates

Polar coordinates are an alternative way to describe locations in a plane, using a distance and an angle. Instead of using horizontal and vertical distances like rectangular coordinates, polar coordinates specify:

• Distance from a fixed point, known as the origin.
• Angle from a fixed direction, usually the positive x-axis.

This system is particularly useful in scenarios where circular or rotational symmetry is present, making calculations more intuitive and straightforward.
To define a point using polar coordinates, we use the notation \((r, \theta)\), where:

• \(r\) is the radial distance from the origin.
• \(\theta\) is the angle measured from the positive x-axis, in radians or degrees.

It's important to understand that the same point can have multiple polar coordinate representations due to the cyclical nature of angles.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a familiar system used to locate points on a plane by referencing two perpendicular axes: typically labeled x (horizontal) and y (vertical). This system is intuitive for many practical applications, like navigation on a grid or working with equations of lines and curves.
In this system, each point is defined by an ordered pair \((x, y)\), where:

• \(x\) is the distance along the horizontal axis.
• \(y\) is the distance along the vertical axis.

Rectangular coordinates provide a straightforward way to add, subtract, and manipulate points through algebraic operations. Understanding the relationship between the x and y values can make it easier to visualize and solve geometric problems.

Conversion Formulas

Converting between polar and rectangular coordinates involves using specific formulas to relate the two systems. These formulas allow us to move from one representation to another while maintaining the same point in space.
For converting polar coordinates \((r, \theta)\) to rectangular \((x, y)\), the formulas are:

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

These expressions come from trigonometric identities linking angles and distances.
Conversely, to convert rectangular coordinates \((x, y)\) to polar \((r, \theta)\), we use:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}{(\frac{y}{x})}\)

Understanding these relationships is crucial for translating problems across coordinate systems, especially for visualizing geometric concepts and simplifying complex equations.