Problem 52 - Precalculus: Functions and Graphs

Question

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. $$r^{2} \cos 2 \theta=-9$$

Step-by-Step Solution

Answer

The Cartesian equation is \(x^2 - y^2 = -9\).

Step 1: Understanding the Polar Equation

The given polar equation is \(r^2 \cos 2\theta = -9\). This equation needs to be converted into Cartesian form involving \(x\) and \(y\). Our goal is to express it in terms of \(x\) and \(y\).

Step 2: Use Polar to Cartesian Conversion

To convert from polar to Cartesian coordinates, we use the identities: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\). Also, \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). Substitute \(\cos \theta = \frac{x}{r}\) and \(\sin \theta = \frac{y}{r}\) into these relations.

Step 3: Simplify Using Double Angle Identity

We have \(\cos 2\theta = \left(\frac{x}{r}\right)^2 - \left(\frac{y}{r}\right)^2 = \frac{x^2 - y^2}{r^2}\). Substitute in the equation \(r^2 \cos 2\theta = -9\) to get \(r^2 \frac{x^2 - y^2}{r^2} = -9\), simplifying to \(x^2 - y^2 = -9\).

Step 4: Cartesian Equation

The Cartesian form of the equation is \(x^2 - y^2 = -9\). This is the equation of a hyperbola centered at the origin with vertices along the \(x\)-axis.

Key Concepts

Cartesian coordinates

Cartesian coordinates are a way of representing points in space using a pair of numbers, \(x\) and \(y\). They are named after the mathematician René Descartes. In a 2D plane, the \(x\)-coordinate indicates the position along the horizontal axis, while the \(y\)-coordinate specifies the position along the vertical axis. This system is widely used because it simplifies the representation and analysis of geometric objects.

In the context of converting from polar to Cartesian coordinates, you will use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). These equations help you translate the radial distance \(r\) and angle \(\theta\) (from the polar coordinate system) into the familiar Cartesian plane. Understanding these basics ensures a smooth conversion between the two systems and allows you to plot points and graphs effectively in \(x\)- extit{y} coordinate space.

Polar coordinates

Polar coordinates offer a different way to represent points, using a distance and an angle. Instead of moving along an \(x\) and \(y\) axis, polar coordinates locate a point based on how far it is from a fixed center point (the origin), known as the radial coordinate \(r\), and the angle \(\theta\) it makes with the positive \(x\)-axis.

This system is especially useful for problems involving circular or rotational symmetry. For example, a circle with radius 5 centered at the origin is easily represented in polar form as \(r = 5\).

• The radial coordinate \(r\) specifies the distance from the origin.
• The angular coordinate \(\theta\) denotes the direction.

To convert polar coordinates to Cartesian coordinates, use the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). This method gives you a comprehensive outlook on how different coordinate systems can be interconnected seamlessly.

Hyperbola

A hyperbola is a type of conic section that looks somewhat like two mirrored curves opposing each other. In the Cartesian coordinate system, the standard equation of a hyperbola is \((x^2/a^2) - (y^2/b^2) = 1\), where \(a\) and \(b\) are real numbers that determine the shape and size.

In our specific case, the equation \(x^2 - y^2 = -9\) resembles a different form of a hyperbola. This modified form implies that the hyperbola is centered at the origin (0,0) and opens along the \(x\)-axis.

• The vertices are situated on the \(x\)-axis.
• The ‘negative’ in front of the \(9\) conforms to this hyperbola being more ‘horizontal’.

Hyperbolas are important in physics and engineering, especially when discussing wave propagation and orbits. Recognizing them quickly on a graph is a key skill developed through exercises like these.

Double angle identity

Double angle identities are a set of trigonometric formulas that express trigonometric functions of double angles, like \(2\theta\), in terms of single angles. A common double angle identity can be expressed for cosine as \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta.\)

Using this identity, you can simplify complex trigonometric equations into more manageable forms. For instance, in converting the polar equation \(r^2 \cos 2\theta = -9\), the double angle identity helps express \(\cos 2\theta\) in terms of \(\cos \theta\) and \(\sin \theta\).

• It provides a bridge from polar expressions to Cartesian equations.
• Aids in expressing \(r^2 \cos 2\theta\) as \(\frac{x^2 - y^2}{r^2}\).

These identities are essential tools in trigonometry, leading to simpler calculations and deeper insights into geometric patterns and transformations.