To convert rectangular coordinates, [tex]\((-1, -2)\)[/tex], into polar coordinates, [tex]\((r, \theta)\)[/tex], follow these steps:
1. Calculate [tex]\( r \)[/tex]:
The radius [tex]\( r \)[/tex] is found using the Pythagorean theorem. It gives the distance from the origin to the point [tex]\((-1, -2)\)[/tex].
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
For the given coordinates [tex]\((-1, -2)\)[/tex]:
[tex]\[
r = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.23606797749979
\][/tex]
2. Calculate [tex]\( \theta \)[/tex]:
The angle [tex]\( \theta \)[/tex], measured in radians, can be found using the arctan (inverse tangent) function. Since both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are negative, which places the point in the third quadrant, we use the arctangent to find the angle relative to the positive x-axis, then adjust based on the quadrant.
[tex]\[
\theta = \arctan\left(\frac{y}{x}\right)
\][/tex]
For [tex]\((-1, -2)\)[/tex]:
[tex]\[
\theta = \arctan\left(\frac{-2}{-1}\right) = \arctan(2)
\][/tex]
However, since the point is in the third quadrant, we need to account for this by adding [tex]\(\pi\)[/tex] to the angle result obtained from [tex]\(\arctan(2)\)[/tex], because the [tex]\(\arctan\)[/tex] function returns values in [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
But directly considering the proper function for all quadrants:
[tex]\(\theta = \arctan2(y, x) = \arctan2(-2, -1) \approx -2.0344439357957027\)[/tex]
3. Combine [tex]\( r \)[/tex] and [tex]\( \theta \)[/tex]:
The polar coordinates are then:
[tex]\[
(-1, -2) = \left( 2.23606797749979, -2.0344439357957027 \right)
\][/tex]
So, the rectangular coordinates [tex]\((-1, -2)\)[/tex] converted to polar coordinates are:
[tex]\[
(-1, -2) = \left(2.23606797749979, -2.0344439357957027\right)
\][/tex]
