Sure! Let's determine which expression is equivalent to [tex]\(\cot \theta\)[/tex] given that [tex]\(\tan \theta = -\frac{3}{8}\)[/tex].
To start off, we need to recall the relationship between [tex]\(\tan \theta\)[/tex] and [tex]\(\cot \theta\)[/tex]. The cotangent is the reciprocal of the tangent, meaning:
[tex]\[
\cot \theta = \frac{1}{\tan \theta}
\][/tex]
Given [tex]\(\tan \theta = -\frac{3}{8}\)[/tex], we can substitute this into the equation to find [tex]\(\cot \theta\)[/tex]:
[tex]\[
\cot \theta = \frac{1}{-\frac{3}{8}} = -\frac{8}{3}
\][/tex]
Now let's evaluate the given options to see which one is equivalent to [tex]\(\cot \theta\)[/tex]:
1. [tex]\(\frac{1}{-\frac{3}{8}}\)[/tex]
This expression also represents [tex]\(\cot \theta\)[/tex], as we just computed:
[tex]\[
\frac{1}{-\frac{3}{8}} = -\frac{8}{3}
\][/tex]
This is indeed equivalent to [tex]\(\cot \theta\)[/tex].
2. [tex]\(\frac{3}{8} + 1\)[/tex]
Let's simplify this expression:
[tex]\[
\frac{3}{8} + 1 = \frac{3}{8} + \frac{8}{8} = \frac{3 + 8}{8} = \frac{11}{8} = 1.375
\][/tex]
This is not equivalent to [tex]\(\cot \theta\)[/tex], since [tex]\(\cot \theta = -\frac{8}{3}\)[/tex].
3. [tex]\(\sqrt{1 + \left(-\frac{8}{3}\right)^2}\)[/tex]
Let's simplify this expression:
[tex]\[
\sqrt{1 + \left(-\frac{8}{3}\right)^2} = \sqrt{1 + \left(\frac{64}{9}\right)} = \sqrt{\frac{9}{9} + \frac{64}{9}} = \sqrt{\frac{73}{9}} = \frac{\sqrt{73}}{3} \approx 2.848
\][/tex]
This is also not equivalent to [tex]\(\cot \theta\)[/tex], since [tex]\(\cot \theta = -\frac{8}{3}\)[/tex].
4. [tex]\(\left(-\frac{3}{8}\right)^2 + 1\)[/tex]
Let's simplify this expression:
[tex]\[
\left(-\frac{3}{8}\right)^2 + 1 = \left(\frac{9}{64}\right) + 1 = \frac{9}{64} + \frac{64}{64} = \frac{9 + 64}{64} = \frac{73}{64} = 1.140625
\][/tex]
This is also not equivalent to [tex]\(\cot \theta\)[/tex].
After evaluating all the options, we see that the expression [tex]\(\frac{1}{-\frac{3}{8}}\)[/tex] is the only one equivalent to [tex]\(\cot \theta\)[/tex].
So, the correct expression equivalent to [tex]\(\cot \theta\)[/tex] is:
[tex]\[
\boxed{\frac{1}{-\frac{3}{8}}}
\][/tex]
