Problem 11 - Precalculus

Question

Convert to rectangular form. $$r=3 \cos 2 \theta$$

Step-by-Step Solution

Answer

The rectangular form is \( x^2 - y^2 = 3 \).

Step 1: Identify polar coordinates and equivalent relations

In polar coordinates, any point can be represented as \[ r = r(\theta), \theta = \theta \text{ (angle)} \]. Moreover, we have relationships between polar and rectangular coordinates for converting: \[ x = r \cos \theta \text{ and } y = r \sin \theta \]. In this exercise, you are given \( r = 3 \cos 2\theta \) as the polar form, and our goal is to convert it to rectangular form.

Step 2: Express the polar equation using double-angle identities

Given the polar form \( r = 3 \cos 2\theta \), make use of the double-angle identity: \[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]Alternatively, we will use \( \cos^2 \theta = \frac{1 + \cos 2\theta}{2} \) and \( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} \) for another approach.Try using these identities to simplify the expression.

Step 3: Substitute identities into the equation

Using the identity, \( \cos 2\theta = 2\cos^2 \theta - 1 \), substitute it back into the equation:\[ r = 3(2\cos^2\theta - 1) = 6\cos^2\theta - 3 \]

Step 4: Substitute using rectangular coordinate relationships

Now, apply the rectangular relationships:\x = r \cos \theta\ and \y = r \sin \theta.\[ \text{We have } \cos^2 \theta = \left( \frac{x}{r} \right)^2 = \frac{x^2}{r^2}. \]Replace \( r \) with the expression found:\[ 6\cos^2 \theta - 3 = r \]\[ i \quad r = 6 \left( \frac{x^2}{r^2} \right) - 3 \]

Step 5: Clear the fractions to express in a form using only x and y

From the previous step:\[ r^3 = 6x^2 - 3r^2 \] Express in terms of \( x \) and \( y \) by multiplying through by \( r \) to eliminate the \( r^3 \): \[ r^2 = x^2 + y^2 \] \[ r(x^2 + y^2) = 6x^2 - 3r^2 \] After multiplying through by \( r \): \[ 6x^2 - 3(x^2 + y^2) = 0 \] which simplifies to: \[ 3x^2 - 3y^2 = 9 \]

Step 6: Solve and simplify the equation

Continuing from the previously simplified equation:\[ 3x^2 - 3y^2 = 9 \] Divide each term by 3:\[ x^2 - y^2 = 3 \].This is the rectangular form of the original polar equation.

Key Concepts

Polar Coordinates

Polar coordinates offer a way to represent points in a plane based on their distance from a reference point and the angle from a reference direction. It uses the notation \((r, \theta)\), where:

• \(r\) is the radial distance from the origin.
• \(\theta\) is the angular coordinate, representing the angle relative to a fixed direction, typically the positive x-axis.

This system is especially useful in contexts involving circular and rotational motion, where angles and distances provide more natural descriptors compared to the rectangular system of \((x, y)\) coordinates. Key conversion formulas link these coordinates to the Cartesian system, allowing one to find \(x = r \cos \theta\) and \(y = r \sin \theta\). Understanding these relationships is fundamental for converting formulas, as demonstrated in this exercise.

Double-Angle Identity

The double-angle identities are special trigonometric formulas that can simplify expressions involving angles, especially those that are doubled. For cosine, the double-angle identity states:\[\cos 2\theta = \cos^2 \theta - \sin^2 \theta\]This identity is derived from the sum formulas of trigonometry, and can also be expressed in alternative forms:

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

These identities allow for flexible rearrangement of expressions. In this specific problem, we used the identity \(\cos 2\theta = 2\cos^2 \theta - 1\) to facilitate the conversion of the polar coordinate expression into its rectangular form. This simplification is crucial when dealing with trigonometric expressions in different coordinate systems.

Polar to Rectangular Conversion

Converting coordinates from polar to rectangular form relies on a set of fundamental equations that relate the two systems. When converting an equation given in polar form, such as \(r = 3 \cos 2\theta\), our goal is to express it in terms of \(x\) and \(y\).

• Utilize the rectangular relationships: \(x = r \cos \theta\) and \(y = r \sin \theta\).
• Solve for \(\cos \theta\) or \(\sin \theta\) in terms of \(x\) and \(y\).
• Substitute these into your polar equation.

Through this process, we eventually transform polar equations to fit within the Cartesian system. This involves applying trigonometric identities, such as the double-angle identities, and using algebraic manipulation to express all terms with the variables \(x\) and \(y\). In this exercise, by following through these steps, the conversion culminated in the equation \(x^2 - y^2 = 3\), which is expressed purely in rectangular terms.

Trigonometric Identities

Trigonometric identities are equations that are true for all angles and are fundamental tools in trigonometry. They are used to simplify expressions and solve equations. Some of the most commonly used identities include:

• Pythagorean Identities: \(\sin^2 \theta + \cos^2 \theta = 1\).
• Angle Sum and Difference Identities.
• Double and Half Angle Identities, such as \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\).

Understanding and applying these identities allows one to manipulate trigonometric expressions comprehensively. They are particularly useful in scenarios involving transformations between polar and rectangular coordinates, where algebraic manipulation helps in expressing equations in the desired form. In this problem, identities play a crucial role in expressing the polar equation \(r = 3 \cos 2\theta\) in a form that can be analyzed in the rectangular coordinate system, simplifying it eventually to produce \(x^2 - y^2 = 3\).