Problem 45 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Cartesian Equation

The given equation is in the Cartesian coordinate system as \( x = 4 \). This represents a vertical line in the plane where the x-coordinate of every point on the line is 4.

Step 2: Recall the Polar Coordinate System

In polar coordinates, points are expressed as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. We use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\) to convert between Cartesian and polar coordinates.

Step 3: Substitute for x in Polar Form

Substitute the expression for \(x\) in terms of polar coordinates into the given equation. So, we write \(r \cos \theta = 4\). This is the equation in polar form.

Key Concepts

Cartesian Coordinates

Cartesian coordinates form a two-dimensional plane made up of the x and y axes. Each point in this system is defined by an ordered pair \( (x, y) \). The x-coordinate tells us the horizontal position while the y-coordinate tells us the vertical position.
Understanding Cartesian coordinates is crucial for graphing equations and visualizing the relationships in algebraic expressions.
For instance, given the equation \( x = 4 \), every point on this vertical line has an x-coordinate of 4, regardless of its y-coordinate. This helps to understand the representation of basic equations in graphical form.

• A helpful way to remember the axes is that the x-axis runs left to right (horizontal), while the y-axis runs up and down (vertical).
• Points like \( (4, y) \) on the line show the verticality by different y values with constant x value.

Polar Form Conversion

Converting Cartesian coordinates to polar form involves a switch from \( (x, y) \) coordinates to \( (r, \theta) \) coordinates, where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.
When converting, we utilize the equations \( x = r \cos \theta \) and \( y = r \sin \theta \).
In our example \( x = 4 \) becomes \( r \cos \theta = 4 \) in polar form. This helps in understanding how points are placed in a circular coordinate system.

• The idea is to reflect each point as a vector originating from the origin, thus emphasizing the radial nature.
• Such conversions are particularly useful in scenarios involving circular and rotational symmetry.

Trigonometric Identities

Trigonometric identities play a critical role when converting between coordinate systems. The key identities that we use for such conversions include:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)

These provide the relationships necessary to transform rectangular Cartesian coordinates into the circular polar coordinates.
Using these identities allows us to reposition points and describe lines, curves, and shapes more effectively in different scenarios.
In the example \( x= 4 \), using \( x = r \cos \theta \) leads to the polar equation \( r \cos \theta = 4 \).
Understanding how trigonometric identities link Cartesian and polar coordinates broadens your ability to solve complex geometry problems.