Question

Describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$ r=\sec \theta $$

Short answer

The graph is a vertical line at \( x = 1 \).

Step-by-step solution

Identify Polar Equation

The given polar equation is \( r = \sec \theta \). We need to describe the graph of this equation.

Rewrite Sec Function

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, the equation becomes \( r = \frac{1}{\cos \theta} \), which implies \( r \cos \theta = 1 \).

Convert to Rectangular Coordinates

In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). Use \( r \cos \theta = 1 \) to find \( x = 1 \). This is the equation of a vertical line at \( x = 1 \) in the rectangular coordinate system.

Describe the Graph

The graph of this equation is a vertical line, which intercepts the x-axis at \( x = 1 \). In polar coordinates, it represents all points for which the product of the radius and the horizontal component is equal to 1.

Key concepts

Polar to Rectangular Conversion

Polar and rectangular coordinates are two ways of representing points in a plane. In polar coordinates, a point is determined by how far away it is from a central point (the radius \( r \)) and the angle \( \theta \) it makes with a reference direction, usually the positive x-axis. When we convert from polar to rectangular coordinates, we use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \). These formulas help translate the polar description of a point to a more familiar Cartesian description using \( x \) and \( y \) coordinates.
For the given polar equation \( r = \sec \theta \), we converted it to a rectangular equation, \( x = 1 \), by recognizing that \( \sec \theta = \frac{1}{\cos \theta} \) and then substituting into the formula \( r \cos \theta = 1 \), resulting in the equation of a vertical line.

Secant Function in Polar Equations

The secant function, \( \sec \theta \), is the reciprocal of the cosine function, \( \cos \theta \). In the context of polar coordinates, interpreting \( r = \sec \theta \) can be a bit tricky at first. This is because \( \sec \theta \), being the reciprocal of cosine, blows up (or becomes undefined) where cosine is zero.
Thus, the equation \( r = \frac{1}{\cos \theta} \) only has valid values where the cosine function does not equal zero. These points are important when considering the possible graph shapes because at certain angles like \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \), the secant function would be undefined, leaving gaps or discontinuities. For \( r \cos \theta = 1 \), it transforms into a straightforward linear equation when viewed in the rectangular system.

Vertical Lines in Cartesian Coordinates

In the rectangular coordinate system, a vertical line is a line that runs up and down the graph, going through a particular point on the x-axis. The simplest equation for a vertical line is \( x = c \), where \( c \) is a constant that signifies the line's crossing point along the x-axis.
Vertical lines are unique because no two points on them will have different x-coordinates; every point shares the same x-value.

• The line has an undefined slope.
• Every point on the line is equidistant horizontally from the origin.
• For vertical lines, it doesn't matter what the value of \( y \) is as the line stretches indefinitely up and down.

Converting the polar equation \( r = \sec \theta \) into \( x = 1 \) represents a typical vertical line where the x-coordinate is fixed at 1, demonstrating the relationship between polar and Cartesian views.