Problem 76 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ 4 x^{2} y=1 $$

Step-by-Step Solution

Answer

The polar form of the equation is \( r = \sqrt[3]{\frac{1}{4 \cos^2(\theta) \sin(\theta)}} \).

Step 1: Identify rectangular coordinate equations

The given equation in rectangular coordinates is \( 4x^2 y = 1 \).

Step 2: Recall the conversion formulas

Recall that the conversions from rectangular coordinates to polar coordinates are: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).

Step 3: Substitute the polar coordinates

Substitute \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) into the given equation. This results in: \( 4 (r \cos(\theta))^2 (r \sin(\theta)) = 1 \).

Step 4: Simplify the equation

Simplify the substituted equation: \( 4 r^2 \cos^2(\theta) \cdot r \sin(\theta) = 1 \). Combine like terms to get \( 4 r^3 \cos^2(\theta) \sin(\theta) = 1 \).

Step 5: Solve for \( r \)

Isolate \( r \) by dividing both sides by \( 4 \cos^2(\theta) \sin(\theta) \): \( r^3 = \frac{1}{4 \cos^2(\theta) \sin(\theta)} \). Take the cube root of both sides to solve for \( r \): \( r = \sqrt[3]{\frac{1}{4 \cos^2(\theta) \sin(\theta)}} \).

Key Concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two values, usually denoted as \(x\) and \(y\), to describe a point on a plane. The \(x\)-coordinate specifies the horizontal position, while the \(y\)-coordinate specifies the vertical position. Together, they form a pair \((x, y)\) that uniquely identifies a location on the plane.
Rectangular coordinates are familiar because they align with our daily experience of moving left-right and up-down. Cartesian systems are used in many fields for plotting points, graphing equations, and analyzing geometric shapes.
In mathematical expressions, functions can be written in terms of \(x\) and \(y\). For example, the given function \(4x^2 y = 1\) is expressed using rectangular coordinates.

coordinate conversion

When converting from rectangular coordinates to polar coordinates, we remodel the point \((x, y)\) into a form that uses the distance from the origin \((r)\) and the angle from the positive \(x\)-axis \((\theta)\). This transformation involves:

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

In the exercise, we started with an equation in rectangular coordinates, \( 4x^2 y = 1 \). In order to convert it, we substituted \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), transforming the equation step-by-step.
The goal is to attain an equation purely in terms of \(r\) and \(\theta\) by applying these conversion formulas.

trigonometric identities

Trigonometric identities are mathematical equations that relate angles to side ratios in right triangles. These identities are very useful in transforming and simplifying expressions involving trigonometric functions. The most common trigonometric identities involve sine and cosine functions:

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

Understanding these identities allows us to manipulate expressions. For instance, in our transformation, we used \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), and then simplified \(4 (r \cos(\theta))^2 (r \sin(\theta)) \).

polar equations

Polar equations use polar coordinates \((r, \theta)\) instead of rectangular coordinates \((x, y)\). In polar form, equations are often simpler to analyze due to the nature of the radial and angular components.
In the given exercise, the rectangular equation \(4x^2 y = 1\) was converted into a polar equation. By substituting \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), the equation became \( 4 r^3 \cos^2(\theta) \sin(\theta) = 1 \).
Simplifying it further led us to \( r = \sqrt[3]{\frac{1}{4 \cos^2(\theta) \sin(\theta)}} \). This form directly relates the radius \(r\) to the angle \(\theta\), giving a clear and concise representation of the relationship between the variables.