Problem 52 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r^{2}=\sin 2 \theta $$

Step-by-Step Solution

Answer

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

Step 1: Understand Polar to Rectangular Conversion

To convert from polar to rectangular coordinates, we use the relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\). Also, recognize that \(\sin 2\theta = 2 \sin \theta \cos \theta\).

Step 2: Substitute and Simplify

We begin with the given equation: \(r^2 = \sin 2\theta\). Substitute \(\sin 2\theta\) using the double angle identity: \(\sin 2\theta = 2 \sin \theta \cos \theta\). So, the equation becomes \(r^2 = 2 \sin \theta \cos \theta\).

Step 3: Relate Sin and Cos to x and y

From the polar-rectangular relationships, \(\sin \theta = \frac{y}{r}\) and \(\cos \theta = \frac{x}{r}\). Substitute these into the equation: \(r^2 = 2 \left(\frac{y}{r}\right) \left(\frac{x}{r}\right)\).

Step 4: Simplify the Equation

Simplify \(r^2 = 2 \frac{xy}{r^2}\) to \(r^4 = 2xy\), since \(r^2 \cdot r^2 = r^4\).

Step 5: Replace r^4 with (x^2 + y^2)^2

Replace \(r^4\) with \((x^2 + y^2)^2\) since \(r^2 = x^2 + y^2\). So, the equation becomes \((x^2 + y^2)^2 = 2xy\).

Step 6: Solution in Rectangular Form

The final equation in rectangular coordinates is \((x^2 + y^2)^2 = 2xy\). This expresses the original polar equation in terms of x and y.

Key Concepts

Double Angle Identity

The Double Angle Identity is a trigonometric formula that helps simplify expressions involving double angles, such as \(2\theta\). This identity is particularly useful in trigonometry when converting polar equations to rectangular coordinates. By using it, complex trigonometric expressions become more manageable.

In our problem, we have the expression \(\sin 2\theta\). The double angle identity for sine is \(\sin 2\theta = 2 \sin\theta \cos\theta\). This identity allows us to express the double angle in terms of the single angles \(\theta\), which are easier to relate to rectangular coordinates like \(x\) and \(y\).

Converting polar to rectangular coordinates ultimately involves breaking down the problem into expressions that involve \(x\) and \(y\), and the double angle identity is a valuable step in achieving this goal. It simplifies \(\sin 2\theta\) so it can be substituted into other formulas related to \(\sin\theta\) and \(\cos\theta\).

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two values, \(x\) and \(y\), to specify locations on a plane. This system is based on a grid, where the values of \(x\) represent the horizontal position, and \(y\) represents the vertical position.

In the context of converting from polar to rectangular coordinates, it's essential to understand the linkage between the two systems. Polar coordinates involve the radius \(r\) and the angle \(\theta\), but these can be translated into \(x\) and \(y\) by using the equations:

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

These equations mean that any point described in polar coordinates can be re-represented in rectangular coordinates, which are often more practical for graphing complex equations. They allow problems to be visualized more straightforwardly as points on a grid rather than needing to work with angles and radii.

Polar Coordinates

Polar coordinates provide a way to describe a point in terms of its distance from a fixed origin, referred to as \(r\), and the angle \(\theta\) from a fixed direction, usually the positive x-axis. This system is particularly effective for problems involving rotations or circular patterns.

In the given exercise, we're working with a polar equation \(r^2 = \sin 2\theta\). The advantage of polar coordinates is that they can simplify some problems, especially those involving circular symmetry. However, when a task focuses on graphing or algebraic manipulations, translating these into rectangular coordinates is necessary.

The basic relationships for converting from polar to rectangular coordinates include:\( \sin\theta = \frac{y}{r} \) and \( \cos\theta = \frac{x}{r} \). By substituting these into the polar equation, we transform it into a composed form \((x^2 + y^2)^2 = 2xy\) that's visually and algebraically useful for further calculations in rectangular space. These transformations are integral to integrating different coordinate systems in mathematics.