91Ó°ÊÓ

Rectangular coordinates are an essential way to depict points on a plane using two values: the x-coordinate and the y-coordinate. These coordinates tell us exactly how far and in which direction a point is located from the origin (where the x-axis and y-axis meet). For example, the point \((-4, 4)\) is described by moving 4 units to the left (because of the negative sign) on the x-axis and 4 units up on the y-axis.
Using rectangular coordinates is very intuitive as it resembles the grid patterns we often see in graphs or maps. This coordinate system is also known as the Cartesian coordinate system, named after the mathematician René Descartes.

Calculating the radius in polar coordinates involves determining the distance from the origin to the given point in rectangular coordinates. This distance (or radius) can be calculated using the Pythagorean theorem. In general, the formula to find the radius \(r\) is:\[r = \sqrt{x^2 + y^2}\]Using the point \((-4, 4)\), we substitute:

• \(x = -4\)
• \(y = 4\)

We then calculate:\[r = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}.\]This tells us that the point is \(4\sqrt{2}\) units away from the origin. Calculating the radius is a crucial step in converting to polar coordinates since it provides a rotational distance that is consistent regardless of the point's location.

Finding the angle \(\theta\) in polar coordinates involves determining the angle a line from the origin to the point makes with the positive x-axis. For this, you use the inverse tangent function, represented as \(\tan^{-1}\). The angle is calculated using the formula:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]By substituting \(x = -4\)and\(y = 4\)for our given point, we calculate:\[\theta = \tan^{-1}\left(\frac{4}{-4}\right) = \tan^{-1}(-1).\]Since our point is located in the second quadrant (where x is negative and y is positive), it is important to adjust the angle accordingly. Typically, \(\tan^{-1}(-1)\) gives \(-\frac{\pi}{4}\), but in the second quadrant, the angle is: \(\pi - \frac{\pi}{4} = \frac{3\pi}{4}\).
Correctly finding \(\theta\) ensures the point's accurate angular direction in polar coordinates.

Trigonometric functions such as sine, cosine, and tangent are vital when working with both rectangular and polar coordinates. They allow us to describe relationships between angles and distances in a right triangle, which is crucial for converting coordinates. In our context:

• \(\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\)

When converting coordinates, the tangent function's inverse helps us find the angle \(\theta\), as \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).This formula links the vertical change (\(y\)) and the horizontal change (\(x\)) to the tangent.
Remember, in trigonometry involving angles, radians, not degrees, are often used. A full circle \(360°\) is equivalent to \(2\pi\)radians. This ensures smoother calculations in mathematical expressions and operations.