91Ó°ÊÓ

Rectangular coordinates are a way to describe the position of a point in a plane using two values: the horizontal and vertical distances from a reference point, known as the origin. This system is laid out like a grid, with the x-axis running left to right and the y-axis running up and down.

When we say a point is located at (-3, 4), it means you move three units to the left and four units up from the origin.

• x-coordinate (-3): Tells us how far to move left or right. Negative means left.
• y-coordinate (4): Tells us how far to move up or down. Positive means up.

This system is very intuitive for plotting and visualizing data, especially in graphing, because it aligns with how we naturally perceive "left," "right," "up," and "down."

Finding the angle in polar coordinates involves determining the direction from the positive x-axis to the line connecting the origin to the point. This angle, known as \(\theta\), gives us the direction of the point in the plane.

The standard method for calculating this angle is using the \(atan2(y, x)\) function, which correctly handles the division and considers the correct quadrant. For our problem, since the point (-3, 4) is in the second quadrant, \(\theta\) is found by

\[\theta = \pi - atan2(4, -3)\]

This calculation ensures \(\theta\) is positive and correctly aligned within the standard angle measurements. The result, approximately 2.2143 radians, shows how far to rotate from the positive x-axis. Understanding these angles helps in fields like navigation and robotics, where precise direction is crucial.

The distance formula is a key concept in transforming rectangular coordinates to polar coordinates. It gives us the radial distance from the origin, represented as \(r\) in polar coordinates, signifying how far a point lies from the origin in the plane.

The formula is:
\[r = \sqrt{x^2 + y^2}\]

For our example point (-3, 4), we plug the coordinates into the formula: \[r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]

This calculation tells us the straight-line distance from (0, 0) to (-3, 4) is 5 units. The distance formula is invaluable for finding the magnitude of displacement, as well as connecting concepts of geometry to algebra in a simple, effective way.