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The limit to evaluate is:

\[
\lim_{\theta \to 0} \frac{\sin(5\cos(\theta))}{3\sec(\theta)}.
\]

### Step-by-step solution:

1. **Simplify the expression**:
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)}.
   \]
   So, the given limit becomes:
   \[
   \lim_{\theta \to 0} \frac{\sin(5\cos(\theta))}{\frac{3}{\cos(\theta)}} = \lim_{\theta \to 0} \frac{\sin(5\cos(\theta)) \cdot \cos(\theta)}{3}.
   \]

2. **Substitute \(\theta = 0\)**:
   At \(\theta = 0\):
   \[
   \cos(0) = 1.
   \]
   So the expression simplifies to:
   \[
   \lim_{\theta \to 0} \frac{\sin(5 \cdot 1) \cdot 1}{3} = \frac{\sin(5)}{3}.
   \]

3. **Final answer**:
   The limit is:
   \[
   \boxed{\frac{\sin(5)}{3}}.
   \]

Would you like a further breakdown of any step or more details?

### 5 Related Questions:
1. What is the general approach to solve limits involving trigonometric functions?
2. How does L'Hopital's Rule apply to limits with indeterminate forms?
3. Can you explain why \(\lim_{\theta \to 0} \cos(\theta) = 1\) and how this affects limits?
4. How can Taylor series expansions help evaluate limits of trigonometric functions?
5. What are some common trigonometric limit forms that students should memorize?

### Tip:
Always check for simple substitutions before considering more advanced techniques like L'Hopital’s rule, especially when trigonometric functions are involved.

Evaluate the limit lim(θ → 0) [sin(5 cos(θ)) / (3 sec(θ))]

Learn how to solve the trigonometric limit problem lim(θ → 0) [sin(5 cos(θ)) / (3 sec(θ))] with a detailed step-by-step solution. This problem covers essential concepts in trigonometry and calculus, including secant and sine functions, substitution, and limit theorems. Suitable for advanced high school or college-level calculus students.

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