Problem 77 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ x=4 $$

Step-by-Step Solution

Answer

The polar coordinate equation is \( r \cos(\theta) = 4 \).

Step 1: Understand the relationship between rectangular coordinates and polar coordinates

Rectangular coordinates \(x,y \) can be converted to polar coordinates \(r, \theta\) using the equations \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Step 2: Substitute the given equation into the polar coordinate form

The given equation is \( x = 4 \). Using the relationship \( x = r \cos(\theta) \), we substitute \( x \) with 4. So, the equation becomes \( 4 = r \cos(\theta) \).

Step 3: Write the final polar coordinate equation

The final polar coordinate form of the given rectangular coordinate equation is \( r \cos(\theta) = 4 \). Since there are no restrictions on \( r \) or \( \theta \), this is the complete transformation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are used to determine the location of a point in a plane using two numerical values: the horizontal coordinate, commonly called 'x', and the vertical coordinate, known as 'y'. These coordinates rely on a grid system where the 'x' axis runs horizontally and the 'y' axis runs vertically. The position of any point can be determined by how far along the 'x' axis and how far up or down the 'y' axis that point is located.

For example, the point (3, 5) means you move 3 units to the right on the 'x' axis and then 5 units up on the 'y' axis. Rectangular coordinates are exceptionally useful for plotting algebraic equations and are extensively used in geometry, physics, engineering, and various other fields.

Coordinate Transformation

Coordinate transformation is the process of converting one set of coordinates to another. In this context, it involves converting between rectangular coordinates (x, y) and polar coordinates \(r, \theta\). The transformation uses trigonometric relationships.

Here’s how you can convert between these two systems:

• From rectangular to polar:
• bulSuitable expressions:
• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)
• Another necessary structure:
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan(\frac{y}{x})\)

\(r\) represents the radial distance from the origin to the point and \(\theta\) is the angle formed between the positive x-axis and the line connecting the origin to the point.

This conversion process is crucial in various practical applications, such as in physics for describing wave patterns, electromagnetism for point charges, and even computer graphics for rotation and scaling tasks.

Equations in Polar Form

Equations in polar form describe the relationship between the radial distance \(r\) and the angle \(\theta\). Unlike rectangular equations that use x and y coordinates, polar equations use trigonometric functions to relate \(r\) and \(\theta\).

For instance, let's revisit the exercise where we had the rectangular equation \(x = 4\). We transformed it into polar form using the substitution \(x = r\cos(\theta)\). This gave us \(4 = r\cos(\theta)\), simplifying to \(r\cos(\theta) = 4\).

This final polar equation represents a set of points (r, \(\theta\)) where the product of the radial distance \(r\) and the cosine of the angle \(\theta\) always equals 4.

Polar equations are particularly useful in scenarios where circular or spiral patterns are present, as they can more conveniently describe these shapes compared to rectangular equations.