Question

Graph the polar function on the given interval. \(r=3 \sec \theta, \quad(-\pi / 2, \pi / 2)\)

Short answer

The polar graph of the function is a vertical line at \(x = 3\) in the Cartesian plane.

Step-by-step solution

Understanding the Function

The function given is a polar function expressed as \( r = 3 \sec \theta \), where \( \theta \) is the angle measured from the polar axis. The \( \sec \theta \) is the secant function, which is the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \).

Identify Domain Restrictions

Since \( \sec \theta \) involves \( 1/\cos \theta \), it is undefined where \( \cos \theta = 0 \). Within the interval \((-\pi/2, \pi/2)\), \( \cos \theta \) is never zero, meaning \( \sec \theta \) is defined throughout this domain.

Sketch the Function

To sketch, note that for \( \theta = 0 \), \( \cos 0 = 1 \) so \( r = 3 \). As \( \theta \) varies from \(-\pi/2\) to \( \pi/2 \), \( \cos \theta \) changes, causing \( r \) to increase indefinitely as \( \theta \) approaches \(-\pi/2 \) or \( \pi/2 \), since \( \cos \theta \rightarrow 0. \) The curve forms a vertical line at \( x = 3 \) in the Cartesian plane, mapped from polar as \( x = r\cos\theta = 3 \).

Converting to Cartesian Coordinates

To confirm our understanding of the polar equation, convert it to Cartesian coordinates. Using \( x = r\cos \theta \) and \( y = r\sin \theta \), substitute \( r = 3 \sec \theta = \frac{3}{\cos \theta} \) into \( x = 3 \), confirming the Cartesian graph is a vertical line at \( x = 3 \). Thus, the polar equation maps onto this vertical line.

Key concepts

Secant Function

The secant function, notated as \( \sec \theta \), is a trigonometric function that is the reciprocal of cosine. It's defined as \( \sec \theta = \frac{1}{\cos \theta} \). This makes the secant function particularly interesting, because it becomes undefined at points where the cosine function is zero.

• In trigonometry, this means the secant function has vertical asymptotes wherever \( \cos \theta \) equals zero.
• Secant is most often used in problems that involve ratios and projections, such as converting or interpreting growth patterns.

In the context of polar coordinates, the value of \( r \) is directly dependent on the secant function. With mathematical functions like \( r = 3 \sec \theta \), any changes in \( \theta \) are directly tied to changes in \( r \). This dependency makes it crucial to understand the behavior and constraints of \( \sec \theta \) before graphing or solving polar equations.

Domain Restrictions

Domain restrictions are limitations on the values a function's input variable can take. For the secant function, \( \sec \theta = \frac{1}{\cos \theta} \), the restriction arises when \( \cos \theta = 0 \) since division by zero is undefined.

• In our given interval \((-\pi/2, \pi/2)\), \( \cos \theta \) never reaches zero. Hence, \( \sec \theta \) is defined everywhere within this range.
• This means there are no places where \( r = 3 \sec \theta \) becomes undefined in this interval. The function \( \sec \theta \) will produce real, calculable values for all given angles \( \theta \) in \((-\pi/2, \pi/2)\).

Understanding domain restrictions is crucial. It informs us where the function behaves normally, and helps identify intervals that need exclusions or special consideration when solving or graphing the function.

Conversion to Cartesian Coordinates

To merge the understanding of polar and Cartesian systems, one often converts between them. For a polar equation \( r = 3 \sec \theta \), we convert to Cartesian coordinates using \( x = r\cos \theta \) and \( y = r\sin \theta \).

• First, use \( r = \frac{3}{\cos \theta} \) to replace \( \sec \theta \).
• Input this into \( x = r\cos \theta = 3 \), simplifying the problem to drawing a vertical line in the Cartesian plane at \( x = 3 \).

Converting coordinates not only provides a check on our work but often simplifies visualization and solution finding. It also grounds abstract polar representations into more familiar forms, reinforcing understanding.

Graphing Polar Equations

Graphing polar equations translates the mathematical relationships defined by the functions into visual form, offering intuitive insights. With \( r = 3 \sec \theta \), this involves plotting how \( r \) (the radius) changes as \( \theta \) varies.

• At \( \theta = 0 \), \( r = 3 \), this is the starting point on the positive x-axis in polar coordinates.
• As \( \theta \) nears the bounds \(-\pi/2\) or \( \pi/2 \), \( r \) increases indefinitely (tending towards infinity), forming a vertical line in Cartesian terms.

The polar graph shows dynamic changes, tracing paths as the equation dictates. By converting and plotting these changes, an image of how the equation operates emerges. Graphing polar equations not only visualizes changes in Radius and Angle but also ties the analysis with geometry and function behavior at larger scales.