Let's evaluate the expression step by step:

We are given the following:

$0.125 \left[ 8 \cos(0^2) + 2 \left( 8 \cos(0.25^2) \right) + 2 \left( 8 \cos(0.5^2) \right) + 2 \left( 8 \cos(0.75^2) \right) + 8 \cos(1^2) \right]$

Step 1: Evaluate the cosines.

• $\cos(0^2) = \cos(0) = 1$
• $\cos(0.25^2) = \cos(0.0625)$ ≈ 0.9980
• $\cos(0.5^2) = \cos(0.25)$ ≈ 0.9689
• $\cos(0.75^2) = \cos(0.5625)$ ≈ 0.8469
• $\cos(1^2) = \cos(1)$ ≈ 0.5403

Step 2: Plug in the values into the expression.

$= 0.125 \left[ 8(1) + 2 \left( 8(0.9980) \right) + 2 \left( 8(0.9689) \right) + 2 \left( 8(0.8469) \right) + 8(0.5403) \right]$

Step 3: Simplify inside the brackets.

$= 0.125 \left[ 8 + 2(7.984) + 2(7.7512) + 2(6.7752) + 4.3224 \right]$

$= 0.125 \left[ 8 + 15.968 + 15.5024 + 13.5504 + 4.3224 \right]$

$= 0.125 \left[ 57.3432 \right]$

Step 4: Multiply by 0.125.

$0.125 \times 57.3432 = 7.1679$

Thus, the value of the expression is approximately 7.1679.

Would you like further details or have any questions?

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Here are some related questions you could explore:

1. How does the cosine function behave at different values of $x$?
2. What are other numerical methods for approximating integrals similar to this summation approach?
3. How can you generalize this sum for other functions, not just cosine?
4. What role does the coefficient 0.125 play in this evaluation?
5. How would this expression change if sine was used instead of cosine?

Tip: Remember, cosine values oscillate between -1 and 1, making them useful for periodic function analysis.