91Ó°ÊÓ

Rectangular coordinates, also known as Cartesian coordinates, are a way to describe locations in a plane using two values: an x-coordinate and a y-coordinate. This coordinate system is named after the mathematician René Descartes. It helps to visualize and study various points by plotting them on a grid. Each point in a rectangular coordinate system is expressed as

• \( (x, y) \),

where

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

For a given point, negative values for \( x \) and \( y \) indicate positions to the left and below the origin (0,0), respectively. Understanding the placement of these coordinates helps in converting them into other forms, like polar coordinates. When converting, it's crucial to first clearly identify the values of \( x \) and \( y \). In our problem, the point \((-\sqrt{5}, -\sqrt{5})\) is situated in the third quadrant of the coordinate plane, where both x and y are negative.

Radius calculation is a fundamental step when converting from rectangular coordinates to polar coordinates. The radius, often denoted as \( r \), represents the distance from the origin to a given point. We use the following formula to calculate \( r \):

• \( r = \sqrt{x^2 + y^2} \)

This equation stems from the Pythagorean theorem, perfect for determining the hypotenuse of a right triangle. Let's break down the calculation for the point \((-\sqrt{5}, -\sqrt{5})\):

• First, square each coordinate: \((-\sqrt{5})^2\) yields 5 for both x and y.
• Add these squared values: 5 + 5 = 10.
• Finally, find the square root of the sum: \( r = \sqrt{10} \).

This result, \( \sqrt{10} \), tells us the point's distance from the origin in the polar coordinate system.

Angle determination is crucial for fully defining a point in polar coordinates. The angle, \( \theta \), describes the direction of the line connecting the origin to the point. It is measured from the positive x-axis towards the point, in a counterclockwise direction, using radians.To find the angle for the point \((-\sqrt{5}, -\sqrt{5})\), the inverse tangent function is useful:

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

In this case:

• \( \theta = \tan^{-1}\left( \frac{-\sqrt{5}}{-\sqrt{5}} \right) = \tan^{-1}(1) = \frac{\pi}{4} \).

Since the original point lies in the third quadrant (where both x and y are negative), we modify \( \theta \) by adding \( \pi \) radians:

• \( \theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \).

This adjustment ensures that the angle accurately reflects the point's quadrant.

Coordinate conversion involves transforming a set of coordinates from one system to another. For our case, this means shifting values from rectangular coordinates (like \((-\sqrt{5}, -\sqrt{5})\)) into polar coordinates. This process is vital for various applications in physics and engineering because it simplifies many problems by switching to a more suitable reference frame.Here's a succinct way to remember the conversion steps:

• Calculate the Radius: Use \( r = \sqrt{x^2 + y^2} \) to compute the distance from the origin.
• Determine the Angle: Utilize \( \theta = \tan^{-1}\left( \frac{y}{x} \right) \) to find the angle, adjusting based on the quadrant placement.
• Combine the Results: The polar coordinates are expressed as \( (r, \theta) \), which for this point are \( (\sqrt{10}, \frac{5\pi}{4}) \).

Overall, converting coordinates is a straightforward yet powerful tool for analyzing spatial problems more intuitively. These steps, when followed meticulously, ensure accurate and consistent conversions.