91Ó°ÊÓ

Polar coordinates are a way of representing points in a plane using a length and an angle.
In this system, each point is described by two values: the radius (r) and the angle (θ).
The radius r is the distance from the point to the origin (center) of the coordinate system. The angle θ is measured from the positive x-axis, moving counterclockwise.
For example, the polar coordinates \((\sqrt{5}, 230^{\text{°}})\) tell us that the point is \(\text{√}5\) units away from the origin and is located at an angle of 230 degrees from the positive x-axis.

Rectangular coordinates, also known as Cartesian coordinates, use the standard x and y axes to locate points in a plane.
Each point is represented by an (x, y) pair, where x is the horizontal distance from the y-axis and y is the vertical distance from the x-axis.
This system is very intuitive, as it directly aligns with our usual way of navigating left-right and up-down directions.
For example, the rectangular coordinates (-1.435, -1.713) tell us the point is 1.435 units to the left and 1.713 units down from the origin.

Converting polar coordinates to rectangular coordinates involves a bit of trigonometry.
The main formulas are: \(\text{x = r \text{·} cos(θ)}\) and \(\text{y = r \text{·} sin(θ)}\).
These formulas come from the relationship between the sides of a right triangle and its angles.
To use these formulas, first convert your angle θ from degrees to radians, since trigonometric calculations are generally done in radians.
In our example, 230° is converted to radians by multiplying by \(\frac{\text{π}}{180}\), which gives us \(\frac{23 \text{π}}{18}\).
Plugging this into our formulas and evaluating the trigonometric functions, we find:

• \(\text{x ≈ \text{√}5 \text{·} cos(230°) ≈ -1.435}\)
• \(\text{y ≈ \text{√}5 \text{·} sin(230°) ≈ -1.713}\)

Thus, we've successfully converted the point from polar to rectangular coordinates.