To determine the polar form of the Cartesian coordinate [tex]\((-1, \sqrt{3})\)[/tex], we need to find both the radius [tex]\(r\)[/tex] and the angle [tex]\(\theta\)[/tex] in polar coordinates.
1. Radius [tex]\(r\)[/tex]:
The radius is the distance from the origin to the point [tex]\((-1, \sqrt{3})\)[/tex]. It can be calculated using the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Given [tex]\(x = -1\)[/tex] and [tex]\(y = \sqrt{3}\)[/tex]:
[tex]\[
r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2
\][/tex]
2. Angle [tex]\(\theta\)[/tex]:
The angle [tex]\(\theta\)[/tex] is the angle formed with the positive [tex]\(x\)[/tex]-axis, measured in the counter-clockwise direction. It can be determined using the arctangent function [tex]\( \text{atan2}(y, x) \)[/tex], which accounts for the signs of the x and y coordinates to determine the correct quadrant:
[tex]\[
\theta = \text{atan2}(y, x)
\][/tex]
Given [tex]\(x = -1\)[/tex] and [tex]\(y = \sqrt{3}\)[/tex]:
[tex]\[
\theta = \text{atan2}(\sqrt{3}, -1)
\][/tex]
This corresponds to calculating the angle where the tangent of the angle is [tex]\(\frac{\sqrt{3}}{-1}\)[/tex]. Evaluating this gives the angle in radians:
[tex]\[
\theta = \frac{2\pi}{3} \approx 2.0944 \text{ radians}
\][/tex]
Therefore, the polar form of the coordinate [tex]\((-1, \sqrt{3})\)[/tex] is:
[tex]\[
(r, \theta) = \left(2, \frac{2\pi}{3}\right)
\][/tex]
To summarize:
[tex]\[
r \approx 2
\][/tex]
[tex]\[
\theta \approx 2.0944 \text{ radians}
\][/tex]
The exact value for [tex]\(r\)[/tex] is 2, and the exact value for [tex]\(\theta\)[/tex] is [tex]\(\frac{2\pi}{3}\)[/tex] radians.
