Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The rectangular form of the equation is \( x^3 + xy^2 - y = 0 \).

Step 1: Recall Polar to Rectangular Formulas

In polar coordinates, the relationship between polar and rectangular coordinates is given by the formulas: \( x = r\cos\theta \) and \( y = r\sin\theta \). Additionally, the radius \( r \) can be expressed in rectangular coordinates as \( r^2 = x^2 + y^2 \), and the tangent of angle \( \theta \) is given by \( \tan\theta = \frac{y}{x} \).

Step 2: Replace \( \tan \theta \) with Rectangular Variable

Using the identity \( \tan\theta = \frac{y}{x} \), substitute \( \tan \theta \) in the given equation: \[ r^2 = \frac{y}{x} \]

Step 3: Express \( r^2 \) in Terms of \( x \) and \( y \)

Since we have \( r^2 = x^2 + y^2 \), replace \( r^2 \) with \( x^2 + y^2 \): \[ x^2 + y^2 = \frac{y}{x} \]

Step 4: Clear the Fraction by Multiplying through by \( x \)

Multiply both sides of the equation by \( x \) to eliminate the fraction: \[ x(x^2 + y^2) = y \]

Step 5: Final Rearrangement

Expand the equation and rearrange to express it entirely in terms of \( x \) and \( y \): \[ x^3 + xy^2 = y \] The equation \( x^3 + xy^2 - y = 0 \) is the rectangular form of the given polar equation.

Key Concepts

Polar Coordinates

Imagine you are out on a lake in a boat, navigating by a compass. Polar coordinates are similar to using a compass to pinpoint a location. They describe a point in space with two pieces of information: the distance from a fixed origin, known as the radius, denoted by \( r \), and the direction or angle \( \theta \), which is the angle from a reference direction, typically the positive x-axis.

Polar Coordinates are represented as \((r, \theta)\). This coordinate system is especially useful when dealing with circular and rotational systems. For example, any point that can be defined with a radius length and a direction is perfect for polar coordinates.

• **Radius (\( r \))**: Distance from the origin to the point.
• **Angle (\( \theta \))**: Measured in radians or degrees from the positive x-axis.

This method of locating points is so beneficial in scenarios where direction and distance are more intuitive than horizontal and vertical distances, such as with the paths of planets or antennas.

Unlike the grid-like system of rectangular coordinates, polar coordinates offer a circular and spiral method of identifying position.

Rectangular Coordinates

Rectangular coordinates are what most of us are familiar with from day-to-day math and graphs. Also known as Cartesian coordinates, they denote a point in space using two perpendicular axes, x and y, creating a grid. Each point is represented by two numbers, \((x, y)\), that measure its horizontal and vertical positions.

Think of it like finding a treasure on a map by counting squares over and up or down from a starting point. Rectangular coordinates are ideal for linear, grid-like contexts like city planning or classroom seating arrangements.

• **x-coordinate (\( x \))**: Horizontal distance from the origin.
• **y-coordinate (\( y \))**: Vertical distance from the origin.

They are the backbone of algebraic and geometric calculations where grid layouts are inherent.

Whereas polar coordinates simplify circular and radial descriptions, rectangular coordinates shine in environments ruled by straight lines and right angles. Both coordinate systems are valuable in their respective scenarios, and understanding conversions between them expands our analytical toolkit.

Coordinate Conversion

Converting from polar to rectangular coordinates is somewhat like switching from looking at a pie from above to viewing it from the side. This conversion lets you translate circular movements into grid movements, giving different insights into the same point.

The conversion process uses simple formulas:

• **From polar to rectangular**: Convert using \( x = r\cos\theta \) and \( y = r\sin\theta \), which effectively project the point onto the x and y axes.
• **From rectangular to polar**: Use \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}(\frac{y}{x}) \) to switch back to directional coordinates.

The conversion system is key in fields like physics, where describing motion in circular paths with straightforward x and y offsets can be incredibly useful.

Learning to convert between these systems is crucial for solving complex problems and provides a deeper understanding of how different mathematical representations relate. Once seamless in conversion, one can effortlessly switch between perspectives.

Trigonometric Identities

Trigonometric identities serve as the bridge in the world of coordinate conversion. They relate angles to ratios of sides in right triangles, enabling conversions between different coordinate systems.

For example, in polar to rectangular conversion, the identities \( \sin\theta = \frac{y}{r} \) and \( \cos\theta = \frac{x}{r} \) are indispensable.

• **Tangent Identity**: \( \tan\theta = \frac{y}{x} \) is essential for converting polar equations since it directly relates angle to side ratios.
• **Pythagorean Identity**: \( \sin^2\theta + \cos^2\theta = 1 \) ensures connections between angular and linear descriptions.

These identities allow us to express circular and angular motion in more universally familiar terms, making them fundamental in mathematics and applied sciences.

Without these identities, finding relationships between different coordinate systems would be much more complex or even impossible. Understanding and using these identities leads to a clearer comprehension of underlying mathematical concepts and their applications.