Problem 159 - Calculus Volume 3

Question

Convert the rectangular equation to polar form and sketch its graph. $$ x^{2}-y^{2}=16 $$

Step-by-Step Solution

Answer

The polar form is \(r = \pm \sqrt{\frac{16}{\cos 2\theta}}\). It represents a hyperbola in polar coordinates.

Step 1: Recall the Relationship Between Cartesian and Polar Coordinates

The polar coordinates \((r, \theta)\) are related to rectangular coordinates \((x, y)\) by the equations \(x = r \cos \theta\) and \(y = r \sin \theta\). To convert an equation from rectangular to polar form, replace these expressions into the equation where \(x\) and \(y\) appear.

Step 2: Substitute Polar Coordinate Equivalents

Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the rectangular equation \(x^2 - y^2 = 16\). This gives:\[(r \cos \theta)^2 - (r \sin \theta)^2 = 16\]

Step 3: Simplify the Equation

Simplify the equation by expanding the squares:\[r^2 \cos^2 \theta - r^2 \sin^2 \theta = 16\]Factor out \(r^2\) from the left side:\[r^2 (\cos^2 \theta - \sin^2 \theta) = 16\]

Step 4: Apply Trigonometric Identity

Recall the trigonometric identity \(\cos^2 \theta - \sin^2 \theta = \cos 2\theta\). Apply this identity to the equation:\[r^2 \cos 2\theta = 16\]

Step 5: Solve for r

Divide both sides by \(\cos 2\theta\) to isolate \(r^2\):\[r^2 = \frac{16}{\cos 2\theta}\]Take the square root of both sides:\[r = \pm \sqrt{\frac{16}{\cos 2\theta}}\]

Step 6: Sketch the Graph

The equation \(r = \pm \sqrt{\frac{16}{\cos 2\theta}}\) shows a hyperbolic curve when converted to polar coordinates. Note that the graph consists of two branches because of the \(\pm\) symbol.Sketching it with polar coordinates involves plotting points for different values of \(\theta\) and connecting the dots to see the hyperbolas that extend along the x-axis.

Key Concepts

Trigonometric Identities

Trigonometric identities are handy tools for transforming and simplifying equations in trigonometry. One such identity is the difference formula:

• \( \cos^2 \theta - \sin^2 \theta = \cos 2\theta \)

This identity is particularly useful in our conversion from rectangular to polar equations. Since the equation involves \( x^2 \) and \( y^2 \,\) substituting them with polar counterparts creates trigonometric functions that, with the help of identities, can be condensed into simpler expressions.
In our exercise, this identity helps to relate the factors of the polar radius \( r \) with a double angle expression, simplifying a complex-looking formula. By becoming familiar with such identities, you can better navigate converting and solving trigonometric equations in different coordinate systems.

Rectangular to Polar Conversion

Converting equations from rectangular (Cartesian) to polar coordinates is an essential process in understanding different dimensions of functions. It involves translating the conventional (x, y) plane into polar form, which is described by radius (r) from the origin and an angle (\(\theta\)) from the positive x-axis.

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

These conversions enable graphing and analysis of equations in a manner that might be more intuitive for circular and spiral shapes. In our example, by replacing \( x \) and \( y \) with their polar equivalences, we've transformed a rectangular equation into a polar one, allowing us to express patterns in terms of radial and angular dynamics rather than grid-based x and y.
This technique is valuable for understanding not just static points, but dynamic systems like rotations, waves or spirals, making polar coordinates a powerful tool in mathematics.

Graphing in Polar Coordinates

Graphing in polar coordinates opens up a world of visualization, capturing complex structures like spirals, roses, and sometimes hyperbolic curves. Unlike Cartesian graphs, where we often plot points based on grid-like intersections of x and y coordinates, polar graphs stem from a central origin, spanning outwards while rotating around the angle \( \theta \).
Visualizing the given equation, \( r = \pm \sqrt{\frac{16}{\cos 2\theta}} \), sketches hyperbolic curves that symmetrically expand from the origin.

• The \( \pm \) sign indicates that there are curves on both sides of the x-axis, each mirroring the other.
• As \( \theta \) changes, tracing out points based on changing radius \( r \) captures this symmetry.

To sketch such graphs, it helps to plot a series of points at increments of \( \theta \) and connect them smoothly. This method not only reveals the structure of the hyperbola but also highlights the intriguing complexity often hidden in a simple-looking equation. Polar plotting brings an aesthetic and functional perspective to analyzing equations in mathematics.