Problem 43 - Algebra and Trigonometry

Question

Convert the equation to polar form. $$ x=y $$

Step-by-Step Solution

Answer

Polar form: \(\theta = \frac{\pi}{4} + n\pi\) for \(n \in \mathbb{Z}\).

Step 1: Understand Cartesian Coordinates

In the Cartesian coordinate system, a point is represented by an ordered pair \((x, y)\). The equation given is \(x = y\), which specifies that any point on the line is equidistant from the x-axis and the y-axis.

Step 2: Convert to Polar Coordinates

Polar coordinates represent a point using the distance from the origin \(r\) and the angle \(\theta\) with respect to the positive x-axis. The relationships between polar and Cartesian coordinates are: \[ x = r \cos(\theta) \quad\text{and}\quad y = r \sin(\theta) \] Substitute these into the equation \(x = y\): \[ r \cos(\theta) = r \sin(\theta) \]

Step 3: Solve for the Angle

Since \(r\) cannot be zero for all points on the line except the origin, divide both sides by \(r\). This simplifies the equation to: \[ \cos(\theta) = \sin(\theta) \] This is equivalent to \(\tan(\theta) = 1\), meaning: \[ \theta = \frac{\pi}{4} + n\pi \] for \(n\in\mathbb{Z}\).

Step 4: Write the Polar Form

The polar form of the line is expressed through \(\theta\) since \(r\) remains a variable representing the distance from the origin to any point on the line. Thus, the polar equation of the line is: \[ \theta = \frac{\pi}{4} + n\pi \]

Key Concepts

Cartesian Coordinates

The concept of Cartesian coordinates is fundamental in mathematics and physics. They allow us to pinpoint the location of a point on a two-dimensional plane. This system uses an ordered pair

• The first number, denoted as x, refers to the horizontal position of a point, i.e., how far left or right the point is from the origin.
• The second number, y, refers to the vertical position, indicating how far up or down the point is from the origin.

For example, in the equation \( x = y \), each point on the line is equally distant from both axes, meaning any point where the horizontal distance equals the vertical distance belongs to this line. These coordinates provide a straightforward, grid-like method of locating points based on perpendicular directions.
This method is intuitive because it aligns closely with a map format, creating a sense of directionality and distance that many find easy to grasp.

Distance from the Origin

Distance from the origin in the context of polar coordinates is represented by \( r \). It measures how far a point is from the point (0,0) on the Cartesian plane. This is crucial for understanding how polar coordinates function, as the value of \( r \) indicates the line along which the point lies in conjunction with the variable \( \theta \).
Imagine extending a string from the origin to the point. The length of that string is precisely the distance from the origin, or \( r \). Whether a point is located closely or far away depends on the magnitude of \( r \). It's usually a positive value, unless stated otherwise, because distances cannot be negative. Remember, the origin is the pivotal point when converting Cartesian coordinates, like with the line \( x = y \). This keeps the location of the line consistent regardless of how the coordinates are described.

Angle with the x-axis

In polar coordinates, the angle \( \theta \) is measured from the positive x-axis to the line connecting the origin and the point. This angle can give you a rotational perspective, indicating the direction where the line extending from the origin points. For our exercise, the angle \( \theta \) is crucial.

• Each position can be described as a rotation from the x-axis, denoted by \( \theta \).
• In the equation \( x = y \), solving \( \tan(\theta) = 1 \) shows \( \theta = \frac{\pi}{4} + n\pi \), marking lines equally angled between axes.

Angles in polar coordinates offer a smooth transition between directions, such as when mapping a line that falls at specific rotational increments like our example of \( \theta = \frac{\pi}{4} + n\pi \). Understanding this aspect facilitates converting complex Cartesian equations into polar forms.

Conversion of Coordinates

Conversion between Cartesian and polar coordinates is an essential skill in math and science. This process involves shifting from the familiar (x, y) format to one based on distance and angles (r, \( \theta \)).

• The formulae used are: \[ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) \]
• By substituting these into expressions, like \( x = y \), we derive \( \cos(\theta) = \sin(\theta) \), leading to our polar form, \( \theta = \frac{\pi}{4} + n\pi \).

This conversion is not merely a mathematical exercise but enriches how geometric relationships are visualized, giving insight into angles and radial distances.
It allows us to explore patterns that might be less apparent in Cartesian plots, like radial symmetry or angular orientations, ultimately expanding the realm of mathematical visualization.