Determine whether the function \( f(x)=\sin \left(\frac{\pi}{2}-x\right) \) is continuous at a closed interval \( [-2,2] \).

To determine whether the function \( f(x) = \sin\left(\frac{\pi}{2} - x\right) \) is continuous over the closed interval \( [-2, 2] \), we can follow these steps:

### Step 1: Analyze the function
We can rewrite the given function using a trigonometric identity. Specifically, the identity for sine is:
\[
\sin\left(\frac{\pi}{2} - x\right) = \cos(x)
\]
Thus, the function simplifies to:
\[
f(x) = \cos(x)
\]

### Step 2: Determine continuity of \( \cos(x) \)
The cosine function \( \cos(x) \) is well-known to be continuous for all real numbers. Since \( \cos(x) \) is continuous on the entire real line, it is certainly continuous on the closed interval \( [-2, 2] \).

### Step 3: Conclusion
Since the function \( f(x) = \cos(x) \) is continuous everywhere, it is also continuous on the interval \( [-2, 2] \).

Thus, the function \( f(x) = \sin\left(\frac{\pi}{2} - x\right) \) is continuous on the closed interval \( [-2, 2] \).

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. What are the general properties of trigonometric functions like sine and cosine in terms of continuity and differentiability?
2. How would you determine if a function is differentiable over a given interval?
3. What is the importance of the intermediate value theorem for continuous functions?
4. How can one analyze the continuity of piecewise-defined functions over an interval?
5. What is the graphical representation of the function \( f(x) = \cos(x) \) over the interval \( [-2, 2] \)?

**Tip:** A function that can be expressed in terms of elementary continuous functions (like sine, cosine, or polynomials) is usually continuous over any closed interval in its domain.

Explore whether the function \( f(x) = \sin(\frac{\pi}{2} - x) \) is continuous over the closed interval \( [-2, 2] \). Understand the trigonometric identity used and how continuity applies to \( \cos(x) \) over this range. Suitable for Grades 11-12.

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