To find [tex]\(\cot \theta\)[/tex] given that [tex]\(\tan \theta = -\frac{3}{8}\)[/tex], we need to recall the relationship between the tangent and cotangent functions. The cotangent function is the reciprocal of the tangent function, which means [tex]\(\cot \theta = \frac{1}{\tan \theta}\)[/tex].
Given [tex]\(\tan \theta = -\frac{3}{8}\)[/tex], we can find [tex]\(\cot \theta\)[/tex] by taking the reciprocal of [tex]\(\tan \theta\)[/tex]:
[tex]\[
\cot \theta = \frac{1}{-\frac{3}{8}}
\][/tex]
To simplify this expression, we realize that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore:
[tex]\[
\frac{1}{-\frac{3}{8}} = 1 \times \left(-\frac{8}{3}\right) = -\frac{8}{3}
\][/tex]
Thus, the expression equivalent to [tex]\(\cot \theta\)[/tex] is [tex]\(\frac{1}{-\frac{3}{8}}\)[/tex].
This matches the first option provided:
[tex]\[
\frac{1}{-\frac{3}{8}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\frac{1}{-\frac{3}{8}}
\][/tex]
