To solve the expression \( s = 5 \sin(\pi \times 1) + 2 \cos(\pi \times 1.1) \), let's evaluate each trigonometric function step by step.

1. **Evaluate** \( \sin(\pi \times 1) \):
   \[
   \sin(\pi) = 0
   \]
   So, \( 5 \sin(\pi \times 1) = 5 \times 0 = 0 \).

2. **Evaluate** \( \cos(\pi \times 1.1) \):
   \[
   \cos(\pi \times 1.1) = \cos(1.1\pi)
   \]
   \( 1.1\pi \) is in the second quadrant, where the cosine is negative. Specifically:
   \[
   \cos(1.1\pi) = -\cos(0.1\pi)
   \]
   The value \( \cos(0.1\pi) \approx \cos(18^\circ) \approx 0.9511 \).

   Thus,
   \[
   \cos(1.1\pi) \approx -0.9511
   \]
   Therefore,
   \[
   2 \cos(\pi \times 1.1) \approx 2 \times -0.9511 = -1.9022
   \]

3. **Combine the results**:
   \[
   s = 0 + (-1.9022) = -1.9022
   \]

So, the value of \( s \) is approximately \( \boxed{-1.9022} \).

Would you like more details on any of the steps? 

Here are some related questions:
1. How do you determine the sign of trigonometric functions in different quadrants?
2. Can you explain the unit circle and how it relates to sine and cosine values?
3. What are the exact trigonometric values for common angles like 0, 30, 45, 60, and 90 degrees?
4. How can you calculate sine and cosine for non-standard angles without a calculator?
5. How do phase shifts affect the graphs of sine and cosine functions?

**Tip:** Understanding the unit circle and the sign of trigonometric functions in each quadrant is essential for solving trigonometric expressions efficiently.

Discover how to solve the trigonometric expression s = 5 sin(π) + 2 cos(1.1π) with a detailed, step-by-step explanation. Learn about trigonometric functions, unit circle properties, and how to evaluate sine and cosine for different angles. Ideal for high school students (Grades 9-12).

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