Problem 35 - Algebra and Trigonometry

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi\). $$ (\sqrt{8}, \sqrt{8}) $$

Step-by-Step Solution

Answer

The polar coordinates are \((4, \frac{\pi}{4})\).

Step 1: Understand the Problem

We need to convert the given rectangular coordinates \((x, y) = (\sqrt{8}, \sqrt{8})\) to polar coordinates \((r, \theta)\) where \(r > 0\) and \(0 \leq \theta < 2\pi\).

Step 2: Calculate the Radius \(r\)

The formula for the radius \(r\) in polar coordinates is given by:\[ r = \sqrt{x^2 + y^2} \]Substituting \(x = \sqrt{8}\) and \(y = \sqrt{8}\), we get:\[ r = \sqrt{(\sqrt{8})^2 + (\sqrt{8})^2} = \sqrt{8 + 8} = \sqrt{16} = 4 \]

Step 3: Calculate the Angle \(\theta\)

The angle \(\theta\) is found using the formula:\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]Here, \(x = \sqrt{8}\) and \(y = \sqrt{8}\), so:\[ \theta = \tan^{-1}\left(\frac{\sqrt{8}}{\sqrt{8}}\right) = \tan^{-1}(1) \]The angle whose tangent is 1 is \(\frac{\pi}{4}\) radians.

Step 4: Determine the Correct Range for \(\theta\)

We found \(\theta = \frac{\pi}{4}\), which already lies within the required range of \(0 \leq \theta < 2\pi\). No adjustment is necessary.

Step 5: Write the Polar Coordinates

Combine the values of \(r\) and \(\theta\) to express the polar coordinates. Thus, the polar coordinates are:\[(r, \theta) = (4, \frac{\pi}{4})\].

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent a point in a two-dimensional plane, often using the horizontal and vertical distances from the origin. These are usually denoted as \(x, y\), where \(x\) represents the distance along the horizontal axis and \(y\) represents the distance along the vertical axis.

This form of coordinate system is also known as the Cartesian coordinate system, named after René Descartes. It’s widely used because of its simplicity in describing the location of a point in a grid-like layout which parallels our everyday understanding, like city block layouts.

In this specific exercise, the point \((x, y) = (\sqrt{8}, \sqrt{8})\) is given in rectangular coordinates. This is a numerical representation that places the point diagonally from the origin. When converting to polar coordinates, we're essentially mapping that point onto a system that uses angles and field radius points from the origin.

Angle Calculation

The angle calculation is a critical step when converting rectangular coordinates to polar coordinates. It determines the direction in which the point lies relative to the positive x-axis.

For this conversion, the formula used is:

• \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)

The angle \(\theta\) corresponds to arctangent of the slope created by the line connecting the origin to the point. In our example, since we have \(x = \sqrt{8}\) and \(y = \sqrt{8}\), it translates into:

• \( \theta = \tan^{-1}(1) \)
• This results in \( \theta = \frac{\pi}{4} \)

This angle means our point lies on the line that is equidistant from both axes, forming a 45-degree angle (or \(\frac{\pi}{4}\) radians) between it and the x-axis. It is vital to ensure the angle is within the range of \(0 \leq \theta < 2\pi\) which in this case it naturally is, requiring no further adjustment.

Radius Calculation

The radius in polar coordinates measures the distance from the origin to the point. It helps position the point on the plane, serving as a measure of how far the point is from the central reference.

To calculate the radius \(r\), the formula used is:

• \( r = \sqrt{x^2 + y^2} \)

In our example with \(x = \sqrt{8}\) and \(y = \sqrt{8}\), you compute:

• \( r = \sqrt{(\sqrt{8})^2 + (\sqrt{8})^2} \)
• This simplifies to \( r = \sqrt{16} \)
• Therefore, \( r = 4 \)

The radius clarifies that the point is 4 units away from the origin, helping to visualize its position in a more circular manner than rectangular coordinates allow. This conversion enhances the understanding of distances and angles, making it useful in various applications like physics, engineering, and even navigation.