Problem 55 - Precalculus with Calculus Previews

Question

Find a rectangular equation that has the same graph as the given polar equation. $$ r^{2}=4 \sin 2 \theta $$

Step-by-Step Solution

Answer

The rectangular equation is \( (x^2 + y^2)^2 = 8xy \).

Step 1: Understand Polar Equation

The given polar equation is \( r^2 = 4 \sin 2\theta \). In this equation, \( r \) represents the radius and \( \theta \) represents the angle in polar coordinates.

Step 2: Use Polar-to-Rectangular Formulas

Recall the formulas for converting between polar and rectangular coordinates. We have \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, keep in mind that \( r^2 = x^2 + y^2 \).

Step 3: Substitute and Identify Trigonometric Identities

Substitute the identity \( \sin 2\theta = 2\sin\theta\cos\theta \). Therefore, the given equation becomes \( r^2 = 4 \cdot 2 \sin\theta \cos\theta = 8 \sin\theta \cos\theta \).

Step 4: Express in Terms of \( x \) and \( y \)

From trigonometric identities, \( \sin\theta = \frac{y}{r} \) and \( \cos\theta = \frac{x}{r} \), so \( \sin\theta \cos\theta = \frac{xy}{r^2} \).

Step 5: Substitute \( r \) and Simplify Equation

Using the relation \( r^2 = x^2 + y^2 \), substitute into our modified equation: \( r^2 = 8 \frac{xy}{r^2} \). Multiply both sides by \( r^2 \) to eliminate \( r^2 \) from the denominator, giving \( r^4 = 8xy \). Since \( r^2 = x^2 + y^2 \), substitute \( (x^2 + y^2)\) for \( r^2 \). Hence, \( (x^2+y^2)^2 = 8xy \).

Step 6: Rectangular Equation Result

The rectangular form of the polar equation \( r^2 = 4 \sin 2\theta \) is \( (x^2 + y^2)^2 = 8xy \). This is the final rectangular equation that represents the same relationship in polar coordinates.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points on a plane using a radius and an angle. Instead of the usual grid-like Cartesian coordinates, where we use an (x, y) framework, polar coordinates use a distance from a reference point and an angle from a reference direction.
The system has two components:

• The radius (\( r \)), which tells how far the point is from the origin of the coordinate system.
• The angle (\( \theta \)) measures the direction of that distance from a fixed direction, usually the positive x-axis.

These coordinates are especially useful for problems involving circular or rotational symmetry, like plotting graphs of equations that are cumbersome in rectangular form. By understanding polar coordinates, you can easily convert them into rectangular coordinates to analyze the same curves on a standard x-y plane.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane by its horizontal and vertical distances from two fixed perpendicular lines, known as axes.

• The x-coordinate measures the horizontal distance from the origin along the x-axis.
• The y-coordinate shows the vertical distance along the y-axis.

This system is intuitive for understanding relationships in horizontal and vertical contexts. To convert from polar to rectangular coordinates, you use:

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

This transformation leverages the trigonometric relationships inherent in right triangles. Understanding this allows you to express curves and shapes drawn in a polar context within the familiar rectangular framework.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides are defined. These identities are powerful tools in mathematics to simplify and transform complex equations. In the context of converting from polar to rectangular coordinates, they play an essential role.
For instance, identities such as \( \sin 2\theta = 2 \sin \theta \cos \theta \) can be used to simplify expressions involving angles that are multiples, which is a common scenario in polar coordinates. These are expressed in terms of sine and cosine, which are directly related to rectangular coordinates through the conversion formulas:

• \( \sin \theta = \frac{y}{r} \)
• \( \cos \theta = \frac{x}{r} \)

This allows you to transform a polar equation, which may seem counterintuitive in the polar form, into a more interpretable rectangular form. Such understanding aids in visualizing and plotting these functions, bridging the gap between different mathematical frameworks.