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!