Problem 45 - Precalculus Mathematics for Calculus

Question

Convert the equation to polar form. $$y=x^{2}$$

Step-by-Step Solution

Answer

The polar form is \( r = \frac{\sin \theta}{\cos^2 \theta} \).

Step 1: Recall the Polar Coordinate Definitions

In polar coordinates, a point is represented by \( (r, \theta) \), where \( r \) is the radial distance from the origin and \( \theta \) is the angle from the positive x-axis. The conversion formulas are \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Substitute Cartesian Coordinates to Polar coordinates

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

Step 3: Simplify the Equation

Rewrite \( (r \cos \theta)^2 \) as \( r^2 \cos^2 \theta \) and set up the equation:\( r \sin \theta = r^2 \cos^2 \theta \).

Step 4: Solve for Radius \( r \)

Factor out \( r \) from the right side:\( r \sin \theta = r^2 \cos^2 \theta \). Divide both sides by \( r \) (assuming \( r eq 0 \)) to obtain:\( r = \frac{\sin \theta}{\cos^2 \theta} \).

Step 5: Final Polar Form

Express \( r \) in terms of \( \theta \) to finalize the polar equation as:\( r = \frac{\sin \theta}{\cos^2 \theta} \).

Key Concepts

conversion formulas

Conversion formulas are essential when switching between Cartesian and polar coordinate systems. In mathematics, the Cartesian coordinate system uses pairs

• \((x, y)\) to define a point in the plane.
• In contrast, the polar coordinate system uses pairs \((r, \theta)\).

Here, \(r\) represents the radial distance from the origin, and \(\theta\) is the angle from the positive x-axis to the point.

The conversion formulas are given as:

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

These are derived from trigonometric relationships in a right triangle, where the hypotenuse is equivalent to \(r\). By using these formulas, you can switch between the coordinate frameworks smoothly and solve complex problems more effectively.

cartesian to polar conversion

Converting equations from Cartesian to polar form involves

• using conversion formulas
• substituting the variables \(x\) and \(y\) in terms of the polar variables \(r\) and \(\theta\).

In this process, you replace \(x\) and \(y\) as follows:

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

To convert the equation \(y = x^2\) to polar form, substitute:

• \( y = r \sin \theta = (r \cos \theta)^2 \)

This results in the equation \( r \sin \theta = r^2 \cos^2 \theta \). By simplifying the equation, you aim to express \(r\) solely in terms of \(\theta\), if possible.

Such conversions help in understanding how equations behave in different coordinate systems, expanding your mathematical toolset.

polar equation

A polar equation expresses a relationship between \(r\) and \(\theta\), describing a curve or line in terms of these polar coordinates. Once you have substituted and simplified the variables, the task is to represent the polar equation as simply as possible.

Starting with the Cartesian equation \(y = x^2\), after conversion, you reach

• \( r \sin \theta = r^2 \cos^2 \theta \)

To isolate \(r\), factor both sides by \(r\) and divide, assuming \(r eq 0\), to find:

• \( r = \frac{\sin \theta}{\cos^2 \theta} \)

This is the polar form of the original equation. Such conversion facilitates exploring curves' characteristics more directly, especially for graphs that are symmetric around the origin in circular patterns.

Polar equations provide insights into geometric properties and are instrumental in fields like physics and engineering, where circular motion is common.