To evaluate the six trigonometric functions for [tex]\(\theta = \frac{3\pi}{2}\)[/tex], we need to analyze each function at this specific angle. Here's a step-by-step solution:
1. Sine Function ([tex]\(\sin \theta\)[/tex]):
[tex]\(\sin\left(\frac{3\pi}{2}\right) = -1\)[/tex]
2. Cosecant Function ([tex]\(\csc \theta\)[/tex]):
[tex]\[
\csc \theta = \frac{1}{\sin \theta} \implies \csc\left(\frac{3\pi}{2}\right) = \frac{1}{-1} = -1
\][/tex]
3. Cosine Function ([tex]\(\cos \theta\)[/tex]):
[tex]\(\cos\left(\frac{3\pi}{2}\right) = 0\)[/tex]
4. Secant Function ([tex]\(\sec \theta\)[/tex]):
[tex]\[
\sec \theta = \frac{1}{\cos \theta} \implies \sec\left(\frac{3\pi}{2}\right) = \frac{1}{0}
\][/tex]
The value is undefined (DNE) because division by zero is not defined.
5. Tangent Function ([tex]\(\tan \theta\)[/tex]):
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} \implies \tan\left(\frac{3\pi}{2}\right) = \frac{-1}{0}
\][/tex]
The value is undefined (DNE) because division by zero is not defined.
6. Cotangent Function ([tex]\(\cot \theta\)[/tex]):
[tex]\[
\cot \theta = \frac{1}{\tan \theta} \implies \cot\left(\frac{3\pi}{2}\right)
\][/tex]
The value is undefined (DNE) since [tex]\(\tan\left(\frac{3\pi}{2}\right)\)[/tex] is undefined.
Based on these evaluations, we can fill in the table as follows:
[tex]\[
\begin{tabular}{|l|l|}
\hline
$\sin \theta=-1$ & $\csc \theta = -1$ \\
\hline
$\cos \theta=0$ & $\sec \theta = \text{DNE}$ \\
\hline
$\tan \theta=\text{DNE}$ & $\cot \theta = \text{DNE}$ \\
\hline
\end{tabular}
\][/tex]
