To find the exact value of [tex]\(\sec \left(-\frac{4 \pi}{3}\right)\)[/tex], we need to follow a series of steps involving trigonometric identities and properties of the unit circle. Let's break this down step-by-step.
### Step 1: Understanding the Angle
The angle given is [tex]\(-\frac{4 \pi}{3}\)[/tex]. This is a negative angle, which means we are rotating clockwise from the positive x-axis.
### Step 2: Determine Equivalent Positive Angle
Angles in trigonometry are periodic with a period of [tex]\(2\pi\)[/tex]. To find a coterminal angle that is positive, we can add [tex]\(2\pi\)[/tex] to [tex]\(-\frac{4 \pi}{3}\)[/tex]:
[tex]\[
-\frac{4 \pi}{3} + 2\pi = -\frac{4 \pi}{3} + \frac{6 \pi}{3} = \frac{2 \pi}{3}
\][/tex]
So, [tex]\(-\frac{4 \pi}{3}\)[/tex] is coterminal with [tex]\(\frac{2 \pi}{3}\)[/tex].
### Step 3: Identify the Cosine of the Angle
The cosine function, [tex]\(\cos(\theta)\)[/tex], is the x-coordinate of the point on the unit circle at that angle. For [tex]\(\frac{2 \pi}{3}\)[/tex]:
- [tex]\(\frac{2\pi}{3}\)[/tex] is in the second quadrant, where cosine is negative.
- The reference angle for [tex]\(\frac{2 \pi}{3}\)[/tex] is [tex]\(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\)[/tex].
The cosine of [tex]\(\frac{\pi}{3}\)[/tex] is [tex]\(\frac{1}{2}\)[/tex]. Hence, the cosine of [tex]\(\frac{2 \pi}{3}\)[/tex] (which is [tex]\(-\frac{1}{2}\)[/tex] due to being in the second quadrant):
[tex]\[
\cos \left(\frac{2 \pi}{3}\right) = -\frac{1}{2}
\][/tex]
### Step 4: Calculate the Secant of the Angle
Secant is the reciprocal of the cosine function:
[tex]\[
\sec \left(\theta\right) = \frac{1}{\cos \left( \theta \right)}
\][/tex]
Applying this to [tex]\(\frac{2\pi}{3}\)[/tex]:
[tex]\[
\sec \left(\frac{2 \pi}{3}\right) = \frac{1}{-\frac{1}{2}} = -2
\][/tex]
Thus, the exact value of [tex]\(\sec \left(-\frac{4 \pi}{3}\right)\)[/tex] is [tex]\(-2\)[/tex].
### Conclusion
The correct answer is:
[tex]\[
\boxed{-2}
\][/tex]
