Problem 46 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ x^{2}-y^{2}=1 $$

Step-by-Step Solution

Answer

The polar equation for the given rectangular graph is \(r = \pm\sec(2\theta)\).

Step 1: Recognize Rectangular and Polar Coordinates

The given rectangular equation is \(x^2 - y^2 = 1\). In rectangular coordinates, equations are given in terms of \(x\) and \(y\), whereas in polar coordinates, equations are expressed in terms of \(r\) and \(\theta\). We will convert this rectangular equation to a polar equation.

Step 2: Convert Rectangular to Polar Coordinates

To convert from rectangular coordinates to polar coordinates, use the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the rectangular equation: \((r \cos \theta)^2 - (r \sin \theta)^2 = 1\).

Step 3: Simplify Polar Equation

Simplify the equation: \(r^2 \cos^2 \theta - r^2 \sin^2 \theta = 1\). Factor out \(r^2\): \(r^2(\cos^2 \theta - \sin^2 \theta) = 1\). Use the trigonometric identity \(\cos^2 \theta - \sin^2 \theta = \cos(2\theta)\) to simplify it further.

Step 4: Finalize Polar Equation

Substitute the trigonometric identity to get \(r^2 \cos(2\theta) = 1\). Solve for \(r^2\) by dividing both sides by \(\cos(2\theta)\), resulting in \(r^2 = \frac{1}{\cos(2\theta)}\).

Step 5: Solve for r

Take the square root of both sides to solve for \(r\): \(r = \pm\sqrt{\frac{1}{\cos(2\theta)}}\). Simplify the expression, resulting in the polar form \(r = \pm\sec(2\theta)\).

Key Concepts

Rectangular to Polar Conversion

Rectangular and polar coordinates are two different ways to represent points on a plane. In rectangular coordinates, a point is identified by its horizontal (\(x\)) and vertical (\(y\)) distances from a reference point, usually the origin.
Polar coordinates, on the other hand, use a distance (\(r\)) from the origin and an angle (\(\theta\)) from the positive x-axis to describe a point.

The conversion between these two systems involves some essential formulas:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

Knowing how to switch between these coordinate systems is crucial for solving problems that involve curves and shapes in geometry. In the conversion process, replace \(x\) and \(y\) in the rectangular equation with their polar equivalents. This conversion allows new forms of equations that can simplify the analysis and graphing of complex equations.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the involved variables. They play a vital role when converting equations from rectangular to polar coordinates.
In the exercise, we used the identity \(\cos^2 \theta - \sin^2 \theta = \cos(2\theta)\), known as a double-angle identity, to simplify the expression in the polar conversion.

Here are some other fundamental trigonometric identities useful in mathematical manipulation:

• Pythagorean Identities, like \(\sin^2 \theta + \cos^2 \theta = 1\)
• Reciprocal Identities, such as \(\sec \theta = \frac{1}{\cos \theta}\)
• Angle Sum and Difference Identities, like \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)

Understanding and utilizing these identities is essential for performing complex simplifications and solving trigonometric equations more effectively. They help to transform and simplify trigonometric expressions, leading to more straightforward solutions.

Graphing Equations

Graphing equations, whether in rectangular or polar form, provides a visual way to understand and interpret the relationships of variables.
In rectangular coordinates, graphs are plotted on a standard \(xy\)-plane with straight lines, parabolas, circles, or other shapes resulting from plotting equations involving \(x\) and \(y\).

In polar coordinates, graphing involves a coordinate system with a radial line and concentric circles. The value of \(r\) determines the distance from the origin (pole), and \(\theta\) determines the angle from the positive x-axis, creating curves with patterns such as spirals or rose curves.

By analyzing polar graphs, you gain insights into the behavior of equations based on angle and radius, different from the straight-line way of interpreting rectangular graphs. This perspective can simplify the understanding of complex dynamics in geometry and calculus. Utilizing both forms of graphs expands the possibilities of visualization and interpretation, aiding in more profound comprehension of mathematical concepts.