To solve this problem, we need to determine the values of [tex]\(\sec \theta\)[/tex] and [tex]\(\cot \theta\)[/tex] given that [tex]\(\csc \theta = \frac{3}{2}\)[/tex] and [tex]\(\sec \theta < 0\)[/tex].
### Step-by-Step Solution:
1. Find [tex]\(\sin \theta\)[/tex]:
The cosecant function [tex]\(\csc \theta\)[/tex] is defined as the reciprocal of the sine function:
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
Given [tex]\(\csc \theta = \frac{3}{2}\)[/tex], we can find [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin \theta = \frac{1}{\csc \theta} = \frac{1}{\frac{3}{2}} = \frac{2}{3}
\][/tex]
2. Find [tex]\(\cos \theta\)[/tex]:
Using the Pythagorean identity, which states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substitute [tex]\(\sin \theta = \frac{2}{3}\)[/tex]:
[tex]\[
\left(\frac{2}{3}\right)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
\frac{4}{9} + \cos^2 \theta = 1
\][/tex]
Solving for [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \frac{4}{9} = \frac{5}{9}
\][/tex]
[tex]\[
\cos \theta = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3}
\][/tex]
Given that [tex]\(\sec \theta < 0\)[/tex], [tex]\(\cos \theta\)[/tex] must be negative:
[tex]\[
\cos \theta = -\frac{\sqrt{5}}{3}
\][/tex]
3. Find [tex]\(\sec \theta\)[/tex]:
The secant function [tex]\(\sec \theta\)[/tex] is defined as the reciprocal of the cosine function:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
Substitute [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex]:
[tex]\[
\sec \theta = \frac{1}{-\frac{\sqrt{5}}{3}} = -\frac{3}{\sqrt{5}} = -\frac{3\sqrt{5}}{5}
\][/tex]
Simplifying, we find:
[tex]\[
\sec \theta \approx -1.3416
\][/tex]
4. Find [tex]\(\cot \theta\)[/tex]:
The cotangent function [tex]\(\cot \theta\)[/tex] is defined as the ratio of the cosine function to the sine function:
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\][/tex]
Substitute [tex]\(\cos \theta = -\frac{\sqrt{5}}{3}\)[/tex] and [tex]\(\sin \theta = \frac{2}{3}\)[/tex]:
[tex]\[
\cot \theta = \frac{-\frac{\sqrt{5}}{3}}{\frac{2}{3}} = -\frac{\sqrt{5}}{2}
\][/tex]
Simplifying, we find:
[tex]\[
\cot \theta \approx -1.1180
\][/tex]
### Final Answer
Given that [tex]\(\csc \theta = \frac{3}{2}\)[/tex] and [tex]\(\sec \theta < 0\)[/tex], we have found:
[tex]\[
\sec \theta \approx -1.3416 \quad \text{and} \quad \cot \theta \approx -1.1180
\][/tex]
