To determine the value of [tex]\(\sec \theta\)[/tex] when [tex]\(\tan \theta = -1\)[/tex] and [tex]\(\frac{3 \pi}{2} < \theta < 2 \pi\)[/tex], follow these steps:
1. Identify the Quadrant:
- The given range [tex]\(\frac{3 \pi}{2} < \theta < 2 \pi\)[/tex] places [tex]\(\theta\)[/tex] in the fourth quadrant.
2. Analyze the tangent function:
- In the fourth quadrant, the tangent function ([tex]\(\tan \theta\)[/tex]) is negative.
- Given [tex]\(\tan \theta = -1\)[/tex], we know the angle where this is true is one where the reference angle corresponds to 45 degrees ([tex]\(\frac{\pi}{4}\)[/tex]).
3. Determine the specific angle:
- An angle in the fourth quadrant that fits this criteria is [tex]\(\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)[/tex].
4. Calculate [tex]\(\cos \theta\)[/tex]:
- For [tex]\(\theta = \frac{7\pi}{4}\)[/tex], the reference angle is [tex]\(\frac{\pi}{4}\)[/tex].
- In the fourth quadrant, [tex]\(\cos \theta\)[/tex] is positive.
- Hence, [tex]\(\cos \left( \frac{7\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}\)[/tex].
5. Determine [tex]\(\sec \theta\)[/tex]:
- The secant function is the reciprocal of the cosine function.
- Therefore, [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex].
6. Apply the value of [tex]\(\cos \theta\)[/tex]:
- [tex]\(\sec \left( \frac{7\pi}{4} \right) = \frac{1}{\cos \left( \frac{7\pi}{4} \right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\)[/tex].
Thus, the value of [tex]\(\sec \theta\)[/tex] is [tex]\(\boxed{\sqrt{2}}\)[/tex].
