Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=4 \sec \theta $$

Step-by-Step Solution

Answer

The Cartesian equation is \( x = 4 \), a vertical line.

Step 1: Convert Polar to Cartesian Coordinates

To convert the polar equation to Cartesian, we use the identities: \[ x = r \cos \theta \] and \[ y = r \sin \theta \]. Since \( \sec \theta = \frac{1}{\cos \theta} \), the given equation becomes \( r = \frac{4}{\cos \theta} \). Multiply both sides by \( \cos \theta \) to get \( r \cos \theta = 4 \), which simplifies to \( x = 4 \).

Step 2: Interpret the Resulting Cartesian Equation

The Cartesian equation \( x = 4 \) represents a vertical line in the Cartesian plane where all points have an x-coordinate of 4. This means there is no variation with respect to \( y \); \( y \) can be any real number.

Step 3: Sketch the Graph in the Polar Plane

To sketch the line in the \( r\theta \)-plane, note that \( r = \frac{4}{\cos \theta} \) is undefined at \( \theta = \frac{\pi}{2} + n\pi \) for integer \( n \), which limits \( \theta \) to values where \( \cos \theta eq 0 \). The graph is a vertical line parallel to the y-axis at \( x = 4 \) in the Cartesian plane, which appears as rays extending radially from the origin, intersecting the point (4, 0) in the polar plane when \( \theta = 0 \) or \( \theta = \pi \).

Key Concepts

Polar Equations

Polar equations represent mathematical relationships using polar coordinates, where each point on a plane is defined by a distance from the origin (denoted by \(r\)) and an angle \(\theta\) from the positive x-axis.
These equations are particularly useful for curves that have symmetry around the origin or are centered at a point other than the origin.

• The angle \(\theta\) is usually measured in radians.
• \(r\) is the radial distance to a point from the origin.

The original problem involves the polar equation \(r = 4 \sec \theta\), which is given in a form that makes direct Cartesian interpretation challenging. Polar equations like this can often be converted to Cartesian coordinates to make graph sketching easier.
This specific equation has Cartesian counterparts that describe vertical lines, aiding in visualization and interpretation of the graph.

Cartesian Coordinates

Cartesian coordinates use \(x\) and \(y\) to define points on a plane in an intuitive and straightforward rectangular grid. Each point is identified by its horizontal and vertical distances from a defined origin.

• The \(x\)-axis runs horizontally.
• The \(y\)-axis runs vertically.

In the exercise, converting the polar equation to a Cartesian equation involves applying identities like \(x = r \cos \theta\) and \(y = r \sin \theta\).
With the given polar equation \(r = 4 \sec \theta\), substituting the identity for \(\sec \theta\) (which is \(1/\cos \theta\)) allows simplification to \(x = 4\).
This result shows that in the Cartesian coordinates, we represent a vertical line where \(x\) is constant at 4, demonstrating a crucial aspect of coordinate conversion: how spatial relationships can differ in representation.

Graph Sketching

Graph sketching involves drawing the graph of an equation according to the type of coordinates being used. For polar equations, the graph is often a curve or a line in the \(r\theta\)-plane. In our exercise, sketching the graph helps to visually understand the relationship each polar equation describes. To sketch the graph of the original polar equation \(r = 4 \sec \theta\):

• Recognize that this equation, once converted, corresponds to a vertical line at \(x = 4\) in the Cartesian plane.
• In a polar graph, this appears as rays emanating straight out from the origin and moving in the direction given by \(\theta\).

This line in the polar system maps back as a line through multiple angles, effectively constructing the line in radial lines pointing directly away from the origin at specified \(r\), adding visual clarity to the mathematical description.

Mathematical Identities

Mathematical identities are formulas or equations that express a mathematical truth and apply universally. In polar to Cartesian conversions, common identities are vital tools.
These include relationships like \(x = r\cos\theta\) and \(y = r\sin\theta\), which enable the translation of coordinates from polar to Cartesian systems.

• \(\sec\theta = \frac{1}{\cos\theta}\) is an identity used in the provided exercise.
• Recognizing identities aids in simplifying equations and converting forms.

Understanding these identities can demystify complex-looking equations by translating them into a more familiar form. This is particularly important when polar expressions might impede straightforward comprehension or graphing. The ability to strategically use these identities allows for effective problem-solving in geometry and trigonometry, central to the exercise on coordinate conversions.