To determine the value of [tex]\(\sec \theta\)[/tex] given [tex]\(\tan \theta = -\frac{4}{3}\)[/tex] and [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex], let’s follow these steps:
1. Understand the given information:
- [tex]\(\tan \theta = -\frac{4}{3}\)[/tex]
- [tex]\(\theta\)[/tex] is in the second quadrant, where the angle lies between [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\pi\)[/tex].
2. Identify the relationship between tangent and secant:
We know the trigonometric identity:
[tex]\[
\sec^2 \theta = 1 + \tan^2 \theta
\][/tex]
3. Calculate [tex]\(\sec^2 \theta\)[/tex]:
- Given [tex]\(\tan \theta = -\frac{4}{3}\)[/tex],
[tex]\[
\tan^2 \theta = \left( -\frac{4}{3} \right)^2 = \frac{16}{9}
\][/tex]
- Apply the identity:
[tex]\[
\sec^2 \theta = 1 + \frac{16}{9} = \frac{9}{9} + \frac{16}{9} = \frac{25}{9}
\][/tex]
4. Determine [tex]\(\sec \theta\)[/tex]:
- Take the square root of both sides to solve for [tex]\(\sec \theta\)[/tex]:
[tex]\[
\sec \theta = \pm \sqrt{\frac{25}{9}} = \pm \frac{5}{3}
\][/tex]
5. Consider the quadrant:
Since [tex]\(\theta\)[/tex] is in the second quadrant ([tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex]), and we know that in the second quadrant, the cosine function is negative, [tex]\(\sec \theta\)[/tex] (which is the reciprocal of [tex]\(\cos \theta\)[/tex]) must also be negative.
6. Select the correct value:
[tex]\[
\sec \theta = -\frac{5}{3}
\][/tex]
Thus, the correct value of [tex]\(\sec \theta\)[/tex] is [tex]\(\boxed{-\frac{5}{3}}\)[/tex].
