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To find the exact value of \(\sec\left(\frac{7\pi}{6}\right)\), we first need to understand the relationship between secant and cosine. The secant function is the reciprocal of the cosine function:

\[
\sec \theta = \frac{1}{\cos \theta}
\]

So, we need to determine the cosine of \(\frac{7\pi}{6}\) first.

### Step 1: Find the Reference Angle
\(\frac{7\pi}{6}\) radians is in the third quadrant because it is slightly more than \(\pi\) radians (\(\pi = \frac{6\pi}{6}\)).

The reference angle is:

\[
\theta_r = \frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}
\]

### Step 2: Determine the Cosine Value in the Third Quadrant
In the third quadrant, the cosine value is negative. The cosine of the reference angle \(\frac{\pi}{6}\) is:

\[
\cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\]

Therefore, in the third quadrant:

\[
\cos \left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}
\]

### Step 3: Find the Secant Value
Now, find the secant by taking the reciprocal of the cosine:

\[
\sec \left(\frac{7\pi}{6}\right) = \frac{1}{\cos \left(\frac{7\pi}{6}\right)} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}
\]

We can rationalize the denominator:

\[
\sec \left(\frac{7\pi}{6}\right) = -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}
\]

### Step 4: Match the Value with the Given Options
The correct answer is:

\[
\boxed{-\frac{2\sqrt{3}}{3}}
\]

Would you like more details or have any further questions? Here are some related questions to consider:

1. How do you determine the reference angle for a given angle in radians?
2. Why is the cosine negative in the third quadrant?
3. What is the relationship between the secant and cosine functions?
4. How do you rationalize a fraction with a square root in the denominator?
5. What is the significance of \(\pi\) in trigonometric calculations?
6. How do you convert from radians to degrees?
7. What would \(\sec(\theta)\) be for other common angles like \(\frac{\pi}{4}\) or \(\frac{\pi}{3}\)?
8. How do you graph \(\sec(\theta)\) on a unit circle?

**Tip:** Always consider the quadrant in which an angle lies to determine the sign of trigonometric functions.

Learn how to find the secant of 7π/6 radians step-by-step with detailed explanations. Understand the relationship between secant and cosine functions. Suitable for high school students studying trigonometry.

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