To solve this problem, let's analyze the information given and the properties of the trigonometric functions involved.
Given:
[tex]\[ \tan(\theta) = -1 \][/tex]
And we need to find:
[tex]\[ \sec(\theta) \][/tex]
for:
[tex]\[ \frac{3\pi}{2} < \theta < 2\pi \][/tex]
1. Determine the quadrant:
The interval \(\frac{3\pi}{2} < \theta < 2\pi\) places \(\theta\) in the fourth quadrant. In the fourth quadrant, the tangent of an angle is negative, which is consistent with \(\tan(\theta) = -1\).
2. Analyze the trigonometric relationships:
We know that:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
Given that \(\tan(\theta) = -1\), this implies:
[tex]\[
\frac{\sin(\theta)}{\cos(\theta)} = -1 \implies \sin(\theta) = -\cos(\theta)
\][/tex]
3. Determine \(\sin(\theta)\) and \(\cos(\theta)\):
Using the Pythagorean identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Let \(\cos(\theta) = x\). Then:
[tex]\[
\sin(\theta) = -x
\][/tex]
Substitute these into the Pythagorean identity:
[tex]\[
(-x)^2 + x^2 = 1 \implies x^2 + x^2 = 1 \implies 2x^2 = 1 \implies x^2 = \frac{1}{2} \implies x = \pm\frac{1}{\sqrt{2}}
\][/tex]
4. Determine the sign of \(\cos(\theta)\):
In the fourth quadrant, \(\cos(\theta)\) is positive. Therefore, we take the positive value:
[tex]\[
\cos(\theta) = \frac{1}{\sqrt{2}}
\][/tex]
5. Compute \(\sec(\theta)\):
Recall that \(\sec(\theta)\) is the reciprocal of \(\cos(\theta)\):
[tex]\[
\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}
\][/tex]
Thus, the value of \(\sec(\theta)\) is:
[tex]\[ \sqrt{2} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\sqrt{2}} \][/tex]
