91Ó°ÊÓ

Rectangular coordinates are also commonly known as Cartesian coordinates. They are a way to pinpoint the exact location of a point on a plane. In this system, each point is determined by a pair of numerical coordinates, which are distances from two fixed perpendicular lines, called axes. These axes intersect at a point called the origin.

In our example with the point (-5, -12), the first number, -5, represents the x-coordinate. This shows how far left or right the point is from the origin along the horizontal axis.
The second number, -12, is the y-coordinate, indicating the distance up or down the point is from the origin along the vertical axis. Therefore, the point (-5, -12) lies in the plane, 5 units to the left and 12 units down from the origin. It's helpful to visualize this on a graph to truly grasp the concept.

The Pythagorean Theorem is instrumental in finding the radius when converting from rectangular to polar coordinates. It's a fundamental principle in geometry that describes the relationship between the sides of a right triangle.

According to the theorem, for a right triangle with sides 'a' and 'b', and hypotenuse 'c', the principle is: \[ c^2 = a^2 + b^2 \].
In the context of our point (-5, -12), think of it as forming a right triangle with the axes. The side lengths are the absolute values of -5 and -12.

• The x-value, 5, is one leg of the triangle.
• The y-value, 12, is the other leg.
• The hypotenuse is the radius 'r'.

The radius can be computed as: \[ r = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]. This radius 'r' is crucial in determining the point's location in polar coordinates.

The arctangent function, often denoted as \( \tan^{-1} \), is used to find angles when converting between coordinate systems. It helps calculate the angle \( \theta \) in polar coordinates by taking the ratio of the y-coordinate to the x-coordinate.

Using the point (-5, -12), we determine the angle by computing: \[ \theta = \tan^{-1}\left(\frac{-12}{-5}\right) = \tan^{-1}\left(\frac{12}{5}\right) \].
This result provides the measure of the angle in radians relative to the positive x-axis in standard position. It's approximately 1.176 radians.

• The arctangent is particularly useful because it directly gives us an angle measurement from the tangent ratio, simplifying the process of angle finding.
• Keep in mind that the arctangent function often returns values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

This detail affects which quadrant the point resides in, necessitating additional adjustments for accuracy, particularly in this example.

The third quadrant of a Cartesian plane is where both x and y values are negative. Points in the third quadrant are tricky because they require additional adjustment when converting to polar coordinates.

• The standard range for polar coordinate angles is between 0 and \(2\pi\).
• When finding the angle using arctangent, the result initially places the angle in the first or fourth quadrant.

To position this correctly, consider the entire plane as 360 degrees or \(2\pi\) radians.

In the given example, \[ \theta = \tan^{-1}\left(\frac{12}{5}\right) + \pi \].
This translates the approximated angle value 1.176 radians to the correct value within the third quadrant: \[ 4.318 \text{ radians} \].
Recognizing that the point lies in the third quadrant ensures that the correct angle is applied, maintaining the integrity and accuracy of the polar coordinate conversion.