Problem 51 - 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. $$r^{2} \cos 2 \theta=1$$

Step-by-Step Solution

Answer

The Cartesian equivalent is \(x^2 - y^2 = 1\), a hyperbola.

Step 1: Express in Rectangular Coordinates

The given polar equation is \( r^2 \cos 2\theta = 1 \). To convert to rectangular coordinates, we start with the identity \( \cos 2\theta = \cos^2\theta - \sin^2\theta \). Also, recall that \( r^2 = x^2 + y^2 \), \( \cos \theta = \frac{x}{r} \), and \( \sin \theta = \frac{y}{r} \).

Step 2: Apply Trigonometric Identities

Substitute \( \cos \theta \) and \( \sin \theta \) into the identity \( \cos 2\theta = (\cos^2 \theta - \sin^2 \theta) \). This gives \((\frac{x^2}{r^2} - \frac{y^2}{r^2})\). Simplifying, we find that \(\cos 2\theta = \frac{x^2 - y^2}{x^2 + y^2}\).

Step 3: Substitute and Simplify

Substitution of the identities into the given equation gives \(r^2 (\frac{x^2 - y^2}{x^2 + y^2}) = 1\). Simplifying this yields the Cartesian equation \(x^2 - y^2 = 1\).

Step 4: Interpret the Equation

The equation \(x^2 - y^2 = 1\) represents a hyperbola in the Cartesian plane. To sketch the graph in polar coordinates, think of converting each \(\theta\) value into a corresponding \(r, \theta\) pair using the original polar form.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental mathematical equations that express relationships between the trigonometric functions, such as sine, cosine, and tangent. These identities are crucial when converting between different coordinate systems or simplifying expressions in trigonometry. The identity we use here is the cosine double angle formula:
\[ \cos 2\theta = \cos^2\theta - \sin^2\theta \]
This formula helps us relate a trigonometric function of a doubled angle to those of a single angle.

• Double Angle Formula: It simplifies calculations when angles are multiplied.
• Helps to convert expressions from polar to rectangular.

When converting a polar equation to rectangular form, knowing and applying these identities allows us to express trigonometric functions in terms of rectangular coordinates \(x\) and \(y\).
Understanding these identities and their derivations can make solving complex trigonometric equations straightforward.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, provide a method to specify each point uniquely in a plane using two numerical values, \(x\) and \(y\).
This system is the foundation of the Cartesian plane, a major tool in geometry and graphing. Each point on this plane is given by a pair \( (x, y) \) denoting horizontal and vertical distances from a reference point known as the origin.

• Corresponds to horizontal and vertical axis visual representation.
• Used for graphing equations like lines, circles, and hyperbolas.
• Allows easy mathematical manipulation and understanding of geometrical concepts.

In converting from polar to rectangular coordinates, equations such as \( r^2 = x^2 + y^2 \) and identities \( \cos\theta = \frac{x}{r} \) and \( \sin\theta = \frac{y}{r} \) make the transition seamless, giving a clearer picture of geometric figures on the graph.

Cartesian Equation

A Cartesian equation is one expressed in terms of Cartesian coordinates \(x\) and \(y\), often resulting from converting a polar equation. In our context, the Cartesian equation \(x^2 - y^2 = 1\) was derived from the polar equation \(r^2 \cos 2\theta = 1\).
This transformation requires understanding the relationships represented by trigonometric identities and substituting values correctly.

• Helps convert complex trigonometric patterns into simpler forms.
• Provides clear visualization on a standard xy-plane.

The equation \(x^2 - y^2 = 1\) is a classic form representing a geometric figure known as a hyperbola. The Cartesian equation simplifies understanding and sketching these curves in the xy-plane.

Hyperbola

A hyperbola is a type of conic section created by the intersection of a plane with both halves of a double cone. It is uniquely characterized by its two distinct branches, each approaching two asymptotes but never touching them.
In a Cartesian coordinate system, the standard equation of a hyperbola centered at the origin with horizontal transverse axis is
\[ x^2 - y^2 = 1 \]

• Consists of two symmetric curves opening either left-right or up-down.
• Defined by vertices, foci, and asymptotes (lines which the curve approaches).
• Reflects certain types of exponential growth or decay relationships in real-world applications.

Understanding hyperbolas involves interpreting their Cartesian equations and recognizing key components like vertices and asymptotes. These play a significant role in fields like physics, engineering, and astronomy.