Question

Convert the rectangular equation to a polar equation. \(x^{2} y=1\)

Short answer

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

Step-by-step solution

Substitute Polar Coordinates

In polar coordinates, we can express the rectangular coordinates as \(x = r\cos\theta\) and \(y = r\sin\theta\). Substitute these into the equation \(x^2 y = 1\), leading to: \((r\cos\theta)^2 (r\sin\theta) = 1\).

Simplify the Equation

Simplify the equation from Step 1: \(r^2 \cos^2\theta \cdot r \sin\theta = 1\). This becomes \(r^3 \cos^2\theta \sin\theta = 1\).

Solve for r

Isolate \(r^3\) by dividing both sides by \(\cos^2\theta \sin\theta\): \[r^3 = \frac{1}{\cos^2\theta \sin\theta}\].

Rewrite the Polar Equation

Take the cube root of both sides to express \(r\): \[r = \sqrt[3]{\frac{1}{\cos^2\theta \sin\theta}}\]. This is the polar form of the original rectangular equation.

Key concepts

Rectangular Equation

A rectangular equation is an algebraic expression that describes a curve in the Cartesian coordinate system using the familiar variables of

• $x$
• $y$

The given rectangular equation is $x^2 y = 1$.
This is a three-dimensional equation that involves both $x$ and $y$, typically meaning that the solution will need to address both variables to satisfy the equation. The different powers of $x$ and $y$ imply that it includes both squaring and a linear relationship, which define how $x$ and $y$ relate to each other. This type of equation requires manipulation to understand how it can represent a shape or structure in a coordinate system.
Understanding rectangular equations forms the foundation before moving into other coordinate systems such as polar coordinates. The transformation from rectangular to polar helps visualize the problem in a circular, radial form.

Coordinate Substitution

To convert a rectangular equation to a polar equation, we use coordinate substitution. This means replacing \(x\) and \(y\) with their polar equivalents:

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

These substitutions transition the problem from Cartesian coordinates to polar coordinates.
In the context of the given equation \(x^2 y = 1\), substituting for \(x\) and \(y\) gives \((r\cos\theta)^2 (r\sin\theta) = 1\).
This method helps to express a polar equation where the relationship between the radius \(r\) and angle \(\theta\) is clear. It is incredibly useful because it adjusts the complex notion of position from a linear, perpendicular perspective to one that involves circles and angles, thereby simplifying some kinds of calculations.

Trigonometric Functions

Trigonometric functions play a vital role in converting between rectangular and polar coordinates. These functions describe relationships based on angles and ratios of side lengths in right triangles.
When you substitute \(x = r\cos\theta\) and \(y = r\sin\theta\), the cosine function represents the adjacent side over the hypotenuse, while the sine function represents the opposite side over the hypotenuse in a right triangle.

• Cosine (\(\cos\theta\)): Is used to calculate the x-coordinate.
• Sine (\(\sin\theta\)): Is used to calculate the y-coordinate.

These functions allow us to write expressions such as \(r^3 \cos^2\theta \sin\theta = 1\). But simplification is possible by separately isolating the terms, eventually allowing us to find \(r = \sqrt[3]{\frac{1}{\cos^2\theta \sin\theta}}\).
Mastering trigonometric functions ensures you'll be able to neatly convert complex rectangular equations into manageable polar forms, making analysis of the problem easier and more insightful.