Question

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

Short answer

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

Step-by-step solution

Understanding Rectangular to Polar Conversion

To convert the given rectangular equation to polar form, we need to use the relationships between rectangular coordinates \(x, y\) and polar coordinates \(r, \theta\). These relationships are given by \(x = r \cos \theta\) and \(y = r \sin \theta\). The task is to replace \(x\) and \(y\) in the given equation with these expressions.

Substitute Polar Coordinates

Start by substituting the polar coordinate expressions into the equation \(x^{2} - y^{2} = 16\). This gives us \((r \cos \theta)^{2} - (r \sin \theta)^{2} = 16\). The equation now becomes \(r^{2} \cos^{2} \theta - r^{2} \sin^{2} \theta = 16\).

Simplify Using Trigonometric Identities

Notice that the expression can be simplified using the trigonometric identity \(\cos^{2} \theta - \sin^{2} \theta = \cos 2\theta\). Therefore, the equation becomes \(r^{2} \cos 2\theta = 16\).

Solve for r

Divide both sides of the equation by \(\cos 2\theta\) to isolate \(r^{2}\), resulting in \(r^{2} = \frac{16}{\cos 2\theta}\). Finally, take the square root of both sides to solve for \(r\), giving \(r = \pm \sqrt{\frac{16}{\cos 2\theta}}\).

Sketch the Graph

To sketch the graph, note that the equation \(r = \pm \sqrt{\frac{16}{\cos 2\theta}}\) represents a hyperbola. In polar coordinates, each value of \(r\) corresponds to certain angles \(\theta\), reflecting the symmetry of the hyperbola around the origin. Plotting various points or considering the shape provides a guideline for sketching this hyperbola.

Key concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when working with polar coordinates. These identities provide the foundation for simplifying and solving equations involving angles. One commonly used identity is:

• \(\cos^2 \theta + \sin^2 \theta = 1\)

This identity relates the sine and cosine functions, ensuring they form a complete circle. For the exercise, we also use the identity:

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

This specific identity helped simplify the equation \(r^2 \cos^2 \theta - r^2 \sin^2 \theta\) to \(r^2 \cos 2\theta\).
Knowing these identities can simplify the process of converting rectangular equations to polar form. They allow you to transform complex trigonometric expressions into more manageable forms. This is especially helpful in solving equations or sketching graphs, where recognizing equivalent transformations can lead to quicker solutions.

Rectangular to Polar Conversion

Converting equations from rectangular to polar form is an essential skill. This involves translating traditional \(x\) and \(y\) coordinates into the circular plane of \(r\) and \(\theta\). The relationships you need to remember are:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r^2 = x^2 + y^2\)

These equations bridge the gap between rectangular and polar coordinates.
In the original exercise, we used these relationships to substitute \(x\) and \(y\) in the equation \(x^2 - y^2 = 16\) with polar equivalents. This leads us to the equation \(r^2 \cos^2 \theta - r^2 \sin^2 \theta = 16\). Transforming coordinate systems can sometimes make equations more understandable or practical to work with.

Graphing

Graphing in polar coordinates can at first seem tricky. However, with practice, it provides a unique perspective, especially for curves such as circles, spirals, and hyperbolas.
The equation \(r = \pm \sqrt{\frac{16}{\cos 2\theta}}\) represents a hyperbola in polar form. To sketch this, you need to consider values of \(\theta\) where the graph is defined, avoiding those where the denominator is zero (\(\cos 2\theta = 0\)).
Evaluate several values of \(\theta\) to find corresponding \(r\) values. This helps plot points that illustrate the curve's shape. When sketching polar graphs, symmetry around the pole (the origin) is often a helpful property. This symmetry can guide your plot, ensuring a balanced and accurate representation of the curves.