Problem 14 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$y=2$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Cartesian Equation

The equation \(y = 2\) is a horizontal line in the Cartesian coordinate system. It represents all points in the plane where the \(y\)-coordinate is equal to 2, regardless of the \(x\)-coordinate.

Step 2: Write the Cartesian Equation in Polar Form

The relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\) is given by the equations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Thus, the equation \(y = 2\) can be expressed in polar coordinates as \(r\sin(\theta) = 2\).

Step 3: The Polar Equation

The polar equation corresponding to the line \(y = 2\) is \(r\sin(\theta) = 2\). This represents all the points where the radial distance projection on the vertical axis is 2.

Key Concepts

Cartesian coordinates

Cartesian coordinates form the most common way to represent points in the plane using a pair of numbers. These numbers, \(x\) and \(y\), are known as the Cartesian coordinates of the point. They are measured along the horizontal and vertical axes respectively.

• The \(x\)-coordinate indicates the position along the horizontal axis.
• The \(y\)-coordinate indicates the position along the vertical axis.

In the context of the given exercise, the equation \(y = 2\) is a line parallel to the x-axis. Every point where the \(y\)-coordinate is 2 fits this equation. It's straightforward—no matter what \(x\) value you choose, as long as \(y = 2\), the point lies on the line.

polar equation conversion

Converting from Cartesian to polar coordinates involves quite an interesting process where we switch from the \(x, y\) system to the \(r, \theta\) system. Here, \(r\) is the radial distance from the origin (0,0), and \(\theta\) is the angle formed with the positive x-axis.
The transformation equations are:

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

Given the Cartesian equation \(y = 2\), we use the transformation \(y = r\sin(\theta)\).
So, this leads to \(r\sin(\theta) = 2\) in polar form. This equation describes all points for which the component of the radial distance along the vertical direction equals 2. Converting between coordinate systems can uncover new insights about the relationships between points.

trigonometric identities

Trigonometric identities are mathematical tools that express one trigonometric function in terms of others. They can greatly assist in coordinate transformations or solving complex problems.
Essential identities to understand include:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

In polar and Cartesian conversions, leveraging these identities can simplify complex expressions or assist in verifying equations. For example, when converting \(y = 2\) to \(r\sin(\theta) = 2\), recognizing \(\sin(\theta)\) as a trigonometric function helps intuitively relate changes in \(\theta\) to observations in the y-coordinate.
Familiarity with these identities enables you to handle a variety of problems and simplifies the conversion process between Cartesian and polar coordinates.