Problem 47 - Precalculus Mathematics for Calculus

Question

Convert the equation to polar form. $$x=4$$

Step-by-Step Solution

Answer

In polar form, the equation is \( r = \frac{4}{\cos(\theta)} \).

Step 1: Understanding the Cartesian Equation

The given equation is in Cartesian form: \( x = 4 \). This means that for any point on this line, the x-coordinate is always 4, regardless of the y-coordinate.

Step 2: Identifying the Invariant Cartesian Coordinate

In Cartesian coordinates, the line \( x = 4 \) is a vertical line parallel to the y-axis, passing through the point \((4, y)\). For any point on this line, \( y \) can take any value.

Step 3: Understanding Polar Coordinates

In polar coordinates, a point is defined by \( (r, \theta) \), where \( r \) is the distance from the origin and \( \theta \) is the angle from the positive x-axis.

Step 4: Finding the Relationship for \( \theta \)

For the vertical line \( x=4 \), the angle \( \theta \) will be either \( \frac{\pi}{2} \) or \( \frac{3\pi}{2} \) because in polar coordinates, a vertical line corresponds to these angles depending on the direction of the line. However, since \( x = 4 \) is positive, we consider the angle as \( \theta = 0 \).

Step 5: Expressing \( x = 4 \) in Polar Coordinates

Convert the given Cartesian equation into polar form using \( x = r\cos(\theta) \). Substitute \( x = 4 \) to get \( 4 = r\cos(\theta) \).

Step 6: Final Polar Equation

From the above equation, we have \( r = \frac{4}{\cos(\theta)} \). Therefore, the equation \( x = 4 \) in polar form is \( r = \frac{4}{\cos(\theta)} \), which represents the same vertical line through polar coordinates.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are one of the most common ways to describe locations in a plane.
They use two values, typically called \(x\) and \(y\). These values tell us how far a point is from two reference lines: the x-axis and the y-axis.

• The x-axis is horizontal, and values increase as you move right.
• The y-axis is vertical, and values increase as you go up.

This system allows precise placement of points in a flat plane. For example, the equation \(x = 4\) represents all points on a line parallel to the y-axis, passing through every point where \(x\) is 4.
This is a simple yet powerful way to describe geometric shapes in a two-dimensional space.

Polar Form Conversion

Converting between Cartesian and polar coordinates can seem tricky at first, but it's based on a simple relationship.
Polar coordinates work differently than Cartesian, using a radius and an angle to pinpoint a location.
You can think of \(r\) as the distance from the origin (0,0), and \(\theta\) as the direction or angle from the positive x-axis.

• To switch from Cartesian to polar, use these formulas: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
• For the equation \(x = 4\), apply \(r = \frac{4}{\cos(\theta)}\) to get its polar form.

This conversion opens up new ways to understand points and shapes, especially those that look complicated in the Cartesian system.

Coordinate Systems

Coordinate systems like Cartesian and polar are tools for understanding positions and relationships in space.
While the Cartesian system is grid-based, the polar system relies on circles and angles.

• In Cartesian coordinates, points are laid out in lines on a plane, using horizontal and vertical measurements.
• Polar coordinates express the same points by focusing on the distance from an origin point and angle from a reference line.

Understanding how to navigate between these systems is crucial for solving many mathematical and engineering problems.
Each system can be beneficial depending on the context, such as polar for circular patterns or Cartesian for rectangular grids.