To find the value of [tex]\(\tan \theta\)[/tex] for an angle [tex]\(\theta\)[/tex] in standard position passing through the point [tex]\((-2, -3)\)[/tex], we use the definition of the tangent function in a coordinate plane. The tangent of an angle [tex]\(\theta\)[/tex] (formed by the terminal side and the x-axis) is given by [tex]\(\tan \theta = \frac{y}{x}\)[/tex], where [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are the coordinates of the point on the terminal side.
Let's apply this to the given point [tex]\((-2, -3)\)[/tex]:
1. Identify the coordinates: the point is [tex]\((x, y) = (-2, -3)\)[/tex].
2. Substitute these coordinates into the tangent formula:
[tex]\[
\tan \theta = \frac{y}{x}
\][/tex]
3. Plug in the values [tex]\(x = -2\)[/tex] and [tex]\(y = -3\)[/tex]:
[tex]\[
\tan \theta = \frac{-3}{-2}
\][/tex]
4. Simplify the expression:
[tex]\[
\tan \theta = \frac{-3}{-2} = \frac{3}{2}
\][/tex]
Thus, the numerical value of [tex]\(\tan \theta\)[/tex] is [tex]\(\frac{3}{2}\)[/tex].
Therefore, the correct answer is:
(2) [tex]\(\frac{3}{2}\)[/tex]
