Problem 20 - Algebra and Trigonometry

Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \cos \theta=4 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x = 4 \), representing a vertical line passing through \( x = 4 \).

Step 1: Understand the given polar equation

The given polar equation is \( r \cos\theta = 4 \). We need to transform this equation to rectangular coordinates.

Step 2: Identify the relationship between polar and rectangular coordinates

Recall that in polar coordinates, \(x = r \cos\theta \) and \(y = r \sin\theta \). We use these relationships to transform the equation.

Step 3: Substitute the rectangular form

Substitute \( x \) for \( r \cos\theta \) in the given equation. This gives us \( x = 4 \).

Step 4: Identify the rectangular equation

The rectangular form of the equation is \( x = 4 \).

Step 5: Graph the equation

The graph of the equation \( x = 4 \) is a vertical line passing through \( x = 4 \).

Key Concepts

polar coordinates

Polar coordinates represent a point in a plane based on its distance from a reference point (called the pole) and its angle relative to a reference direction (usually the positive x-axis). In polar coordinates, a point is denoted as \( (r, \theta) \), where:

• r is the radial distance from the pole.
• \( \theta \) is the angle measured counterclockwise from the reference direction.

This system is especially useful for graphing circles and spirals. For instance, the equation \( r = 2 \) would represent a circle of radius 2 centered at the origin.

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes (x and y) to define a point in a plane. A point is given as \( (x, y) \), where:

• x represents the horizontal distance from the origin.
• y represents the vertical distance from the origin.

This system is very straightforward for plotting linear equations and parabolas. In our exercise, we used the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \) to convert from polar to rectangular coordinates.

graphing equations

Graphing equations involves plotting points on a coordinate plane that satisfy a given equation. Each type of coordinates has its own method of graphing:

• For polar coordinates, plot points based on distance and angle from the origin.
• For rectangular coordinates, plot points based on x and y values.

In our example, the given polar equation \( r \cos \theta = 4 \) was converted to the rectangular form \( x = 4 \). This simpler form clearly indicates that the graph is a vertical line passing through \( x = 4 \). Understanding both systems helps in accurately interpreting and graphing various mathematical equations.