Problem 97 - Precalculus

Question

The use of a different system of coordinates simplifies many mathematical expressions and some calculations are performed in an easier way. The rectangular coordinates \((x, y)\) are transformed into polar coordinates by the equations $$x=r \cos \theta \quad \text { and } \quad y=r \sin \theta$$ where \(r=\sqrt{x^{2}+y^{2}}(r \neq 0)\) and \(\tan \theta=\frac{y}{x}(x \neq 0) .\) In polar coordinates, the equation of the unit circle \(x^{2}+y^{2}=1\) is just \(r=1\) In calculus, we use polar coordinates extensively. Transform the rectangular equation to polar form. $$y^{2}-2 y=-x^{2}$$

Step-by-Step Solution

Answer

The polar form of the equation is \( r = 2 \sin \theta \).

Step 1: Rewrite the Equation

Start with the given rectangular equation \[ y^2 - 2y = -x^2 \]Rearrange the terms to make it easier to transform:\[ y^2 - 2y + x^2 = 0 \]

Step 2: Identify Polar Coordinate Equations

Recall that the transformation from rectangular to polar coordinates is given by:\[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \] \[ r = \sqrt{x^2 + y^2} \quad \text{and} \quad \tan \theta = \frac{y}{x} \]

Step 3: Substitute Polar Formulas

Substitute the rectangular coordinates with their polar counterparts in the rearranged equation:\[ (r \sin \theta)^2 - 2(r \sin \theta) + (r \cos \theta)^2 = 0 \]

Step 4: Simplify the Expression

Simplify the expression after substitution:\[ r^2 \sin^2 \theta - 2r \sin \theta + r^2 \cos^2 \theta = 0 \]Notice that \( r^2 \sin^2 \theta + r^2 \cos^2 \theta = r^2(\sin^2 \theta + \cos^2 \theta) = r^2 \), using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \).

Step 5: Finalize the Polar Equation

Factor and simplify:\[ r^2 - 2r \sin \theta = 0 \]Factor out \(r\) (assuming \(r eq 0\)): \[ r(r - 2 \sin \theta) = 0 \]Since \(r eq 0\), the equation is:\[ r = 2 \sin \theta \]

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to pinpoint the location of points in a two-dimensional plane. It's like a map where each point is defined by a pair of numbers, \((x, y)\).

This system is very intuitive as it relates to our everyday understanding of position:

• \(x\) represents the horizontal position.
• \(y\) represents the vertical position.

Think of coordinates as a pair referencing distance from the origin.
Rectangular coordinates are perfect for straightforward, grid-like positioning but sometimes complex problems make it a bit challenging.
In such cases, it's helpful to use a different system, like polar coordinates, where calculations can become simpler.

Polar Transformation

A polar transformation involves converting rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), where \(r\) is the radius, and \(\theta\) is the angle.
Here's how the conversion works:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r = \sqrt{x^2 + y^2}\) gives the distance from the origin to the point.
• \(\tan \theta = \frac{y}{x}\) to find the angle with the positive \(x\)-axis.

This transformation allows us to represent circular or rotational symmetries with ease.
Understanding these relationships can help convert complex algebraic equations into simpler expressions in polar form.

Unit Circle

The unit circle is a fundamental concept in both trigonometry and polar coordinates.
It is a circle centered at the origin \((0, 0)\) with a radius of 1.
In polar coordinates, the equation of the unit circle simplifies to \(r = 1\), meaning each point on the circle is exactly one unit away from the center.

The unit circle is invaluable because:

• It simplifies many trigonometric functions and expressions, making calculations easier.
• It serves as a basic building block for understanding more complex circles and oscillatory motion.

With polar transformation, the equation \(x^2 + y^2 = 1\) turns into \(r = 1\), demonstrating the elegance and utility of polar coordinates for circular paths.