Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=3 \sec \theta$$

Step-by-Step Solution

Answer

The rectangular equation of the polar equation \(r=3 \sec \theta\) is \(x=3\). This equation represents a vertical line passing through the point (3, 0).

Step 1: Understanding the Polar Equation

Our polar equation is \( r = 3 \sec \theta \). This means that the radius 'r' is three times the secant of the angle 'theta'. The secant of an angle is the reciprocal of the cosine of that angle. Therefore, this equation can be rewritten as \( r = \frac{3}{\cos \theta} \).

Step 2: Converting to Rectangular Coordinates

The relationship between polar coordinates \((r, \theta)\) and rectangular coordinates \((x, y)\) are given by the equations, \(x = r \cos \theta\) and \(y= r \sin \theta\). Since we have \( r = \frac{3}{\cos \theta} \), substituting this into the \(x = r \cos \theta\) equation we get \(x = \frac{3}{\cos \theta} \times \cos \theta\), which simplifies to \(x = 3\).

Step 3: Sketch the Rectangular Equation

The rectangular equation \(x = 3\) represents a vertical line through the point (3, 0) on the coordinate plane. Draw such a line on the graph.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are the most common way to express the position of a point on a plane using two numbers: the x-coordinate and the y-coordinate. It works just like a map, where you might move a certain distance east (x direction) and then a certain distance north (y direction) to reach a precise location.

These coordinates are part of the Cartesian coordinate system, which is widely used in algebra and geometry. To convert from polar coordinates to rectangular coordinates, use these relationships:

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

Here, 'r' is the distance from the origin, and 'θ' is the angle measured from the positive x-axis. Understanding these concepts is crucial for tackling problems involving coordinate conversion, as shown in the original exercise where the polar equation \( r = 3 \sec \theta \) converts to the rectangular equation \( x = 3 \). This describes a line parallel to the y-axis—a simple vertical line in the Cartesian plane.

Graph Sketching

Graph sketching is the art of illustrating mathematical equations visually to better understand their behavior on a plane. This involves translating an equation into a pictorial form. For example, the equation \( x = 3 \) is a linear equation representing a vertical line in the Cartesian coordinate system.

To sketch this graph, you would:

• Identify the x-value the line corresponds to—in this case, \( x = 3 \).
• Recognize that for all values of y (the vertical component), the x-component remains constant at 3.
• Draw a straight line passing through all points where x equals 3, parallel to the y-axis.

Graph sketching helps you visualize mathematical outcomes without having to rely solely on numerical values. In polar coordinates, the equation \( r = 3 \sec \theta \) signifies a line due to its conversion to this fixed x-coordinate in rectangular form.

Coordinate Conversion

Coordinate conversion is the process of translating coordinates from one system to another. In mathematics, this often happens between polar and rectangular coordinate systems. The conversion relies on trigonometric relationships.
For a point described in polar coordinates \((r, \theta)\):

• - Convert to rectangular by using \( x = r \cos \theta \) and \( y = r \sin \theta \).
• - To convert back from rectangular to polar, use \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

In our original problem, we had a polar equation \( r = 3 \sec \theta \). By expressing \( \sec \theta \) as \( \frac{1}{\cos \theta} \), we translated this into a rectangular form where x equals a constant value 3, a simple yet powerful example of conversion.