Let's solve the given expression step by step:

\[
(-12 \sin(\pi/6)) + 11 \sec^2(\pi/6)
\]

### Step 1: Simplify \(\sin(\pi/6)\)
\[
\sin(\pi/6) = \frac{1}{2}
\]
So, 
\[
-12 \sin(\pi/6) = -12 \times \frac{1}{2} = -6
\]

### Step 2: Simplify \(\sec(\pi/6)\)
\(\sec(\theta)\) is the reciprocal of \(\cos(\theta)\). Thus:
\[
\sec(\pi/6) = \frac{1}{\cos(\pi/6)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}
\]
Now, square it:
\[
\sec^2(\pi/6) = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}
\]

### Step 3: Simplify \(11 \sec^2(\pi/6)\)
\[
11 \sec^2(\pi/6) = 11 \times \frac{4}{3} = \frac{44}{3}
\]

### Step 4: Combine the results
Now, we combine both terms:
\[
-6 + \frac{44}{3}
\]

To add these, express \(-6\) with a denominator of 3:
\[
-6 = \frac{-18}{3}
\]

So:
\[
\frac{-18}{3} + \frac{44}{3} = \frac{44 - 18}{3} = \frac{26}{3}
\]

### Final Answer:
\[
\frac{26}{3}
\]

Let me know if you would like more details or explanations!

Here are 5 related questions to extend your learning:
1. How would the expression change if \(\pi/6\) was replaced with \(\pi/4\)?
2. What are the values of \(\sec(\theta)\) and \(\sin(\theta)\) for \(\theta = \pi/3\)?
3. How would you solve \((-5 \cos(\pi/3)) + 4 \tan^2(\pi/6)\)?
4. What is the general method for calculating trigonometric values of angles like \(\pi/6\)?
5. How can you apply the concept of secant and sine to real-life problems?

**Tip:** Trigonometric identities like \(\sec(\theta) = \frac{1}{\cos(\theta)}\) can often simplify complex expressions!

Step-by-step solution to the trigonometric expression (-12sin(pi/6)) + 11sec^2(pi/6). Learn how to simplify sine and secant values and combine the results for a final answer of 26/3. Suitable for high school students learning trigonometry.

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