Sure, let's go through the detailed steps to find the value of [tex]\(\sec \theta\)[/tex] given that [tex]\(\tan \theta = -1\)[/tex] and [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex].
Step 1: Identify the Quadrant
Given the range [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex], we're working with angles in the fourth quadrant (the interval from [tex]\(270^\circ\)[/tex] to [tex]\(360^\circ\)[/tex]).
Step 2: Determine the Angle
Knowing that [tex]\(\tan \theta = -1\)[/tex], we recognize that tangent is negative in the fourth quadrant. The reference angle where [tex]\(\tan = 1\)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex] (or [tex]\(45^\circ\)[/tex]). For [tex]\(\tan \theta = -1\)[/tex] in the fourth quadrant, [tex]\(\theta\)[/tex] must be [tex]\((2\pi - \frac{\pi}{4}) = \frac{7\pi}{4}\)[/tex].
So, [tex]\(\theta = \frac{7\pi}{4}\)[/tex].
Step 3: Calculate [tex]\(\sec \theta\)[/tex]
Recall that [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex].
First, we need to find [tex]\(\cos \theta\)[/tex]:
[tex]\[
\theta = \frac{7\pi}{4}
\][/tex]
Since [tex]\(\cos \theta\)[/tex] has the same value as [tex]\(\cos\)[/tex] of its reference angle (but with an appropriate sign for the quadrant), and cosine is positive in the fourth quadrant:
[tex]\[
\cos \left(\frac{7\pi}{4}\right) = \cos \left(2\pi - \frac{\pi}{4}\right) = \cos \left(-\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
Thus,
[tex]\[
\sec \left( \frac{7\pi}{4} \right) = \frac{1}{\cos \left( \frac{7\pi}{4} \right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\][/tex]
Therefore, the value of [tex]\(\sec \theta\)[/tex] given that [tex]\(\tan \theta = -1\)[/tex] and [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex] is:
[tex]\[
\boxed{\sqrt{2}}
\][/tex]
