To convert a point from Cartesian coordinates [tex]\((x, y)\)[/tex] to polar coordinates [tex]\((r, \theta)\)[/tex], we need to find the radius [tex]\(r\)[/tex] and the angle [tex]\(\theta\)[/tex].
1. Calculate the radius [tex]\(r\)[/tex]:
The radius [tex]\(r\)[/tex] is the distance from the origin to the point [tex]\((2, -2)\)[/tex]. This can be found using the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
For the given point [tex]\((2, -2)\)[/tex]:
[tex]\[
r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2.83
\][/tex]
2. Calculate the angle [tex]\(\theta\)[/tex]:
The angle [tex]\(\theta\)[/tex] is the angle between the positive [tex]\(x\)[/tex]-axis and the line segment drawn from the origin to the point [tex]\((2, -2)\)[/tex]. This can be found using the arctangent function, taking into account the signs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
\theta = \tan^{-1}\left(\frac{y}{x}\right)
\][/tex]
For our point [tex]\((2, -2)\)[/tex]:
[tex]\[
\theta = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1)
\][/tex]
The value in radians is approximately:
[tex]\[
\theta = -0.79
\][/tex]
Therefore, the point [tex]\((2, -2)\)[/tex] in Cartesian coordinates converts to approximately [tex]\((2.83, -0.79)\)[/tex] in polar coordinates, with the angle measured in radians.
