To solve the expression $\sec\left(\frac{\pi}{3}\right) + 4 \csc\left(\frac{\pi}{6}\right)$, let’s evaluate each trigonometric function separately.

Step 1: Evaluate $\sec\left(\frac{\pi}{3}\right)$

1. Recall that $\sec(x) = \frac{1}{\cos(x)}$.
2. We know that $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.
3. Therefore, $\sec\left(\frac{\pi}{3}\right) = \frac{1}{\frac{1}{2}} = 2$.

So, $\sec\left(\frac{\pi}{3}\right) = 2$.

Step 2: Evaluate $\csc\left(\frac{\pi}{6}\right)$

1. Recall that $\csc(x) = \frac{1}{\sin(x)}$.
2. We know that $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.
3. Therefore, $\csc\left(\frac{\pi}{6}\right) = \frac{1}{\frac{1}{2}} = 2$.

So, $\csc\left(\frac{\pi}{6}\right) = 2$.

Step 3: Substitute and Simplify

Now we can substitute these values back into the expression: $\sec\left(\frac{\pi}{3}\right) + 4 \csc\left(\frac{\pi}{6}\right) = 2 + 4 \cdot 2 = 2 + 8 = 10$

Final Answer

$\sec\left(\frac{\pi}{3}\right) + 4 \csc\left(\frac{\pi}{6}\right) = 10$

Let me know if you’d like more details or have any questions.

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1. What is the secant function, and how is it related to cosine?
2. How does the cosecant function relate to sine?
3. Why is $\frac{\pi}{3}$ associated with certain trigonometric values?
4. How can similar steps be applied to other trigonometric functions?
5. What are exact trigonometric values, and why do they matter?

Tip: Memorizing key trigonometric values for standard angles can greatly speed up calculations without a calculator.