To find the Taylor series for the function $f(x) = \sin(x)$ centered at $a = \pi$, we'll use the Taylor series expansion formula:

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$

Step 1: Calculate the derivatives of $f(x) = \sin(x)$

Let's find the first few derivatives of $f(x)$:

$f(x) = \sin(x)$ $f'(x) = \cos(x)$ $f''(x) = -\sin(x)$ $f'''(x) = -\cos(x)$ $f^{(4)}(x) = \sin(x)$

Step 2: Evaluate the derivatives at $x = \pi$

Now, we'll evaluate each derivative at $x = \pi$:

$f(\pi) = \sin(\pi) = 0$ $f'(\pi) = \cos(\pi) = -1$ $f''(\pi) = -\sin(\pi) = 0$ $f'''(\pi) = -\cos(\pi) = 1$ $f^{(4)}(\pi) = \sin(\pi) = 0$

Step 3: Construct the Taylor series

Using these values, the Taylor series expansion of $\sin(x)$ around $\pi$ is:

$\sin(x) = f(\pi) + f'(\pi)(x - \pi) + \frac{f''(\pi)}{2!}(x - \pi)^2 + \frac{f'''(\pi)}{3!}(x - \pi)^3 + \cdots$

Substituting the values we found:

$\sin(x) = 0 - 1 \cdot (x - \pi) + 0 \cdot (x - \pi)^2 + \frac{1}{6}(x - \pi)^3 + \cdots$

Simplifying, we get:

$\sin(x) = -(x - \pi) + \frac{(x - \pi)^3}{6} + \cdots$

Final Answer:

The Taylor series for $\sin(x)$ centered at $a = \pi$ is:

$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{(2n+1)!} (x - \pi)^{2n+1}$

This series represents the sine function in the vicinity of $\pi$.

Would you like more details, or do you have any questions?

---

Here are some related questions you might be interested in:

1. How do you determine the radius of convergence for the Taylor series?
2. What happens if we center the Taylor series at a different point, say $a = 0$?
3. How does the Taylor series for $\cos(x)$ differ when centered at $\pi$?
4. Can you derive the Taylor series for $\sin(x)$ using a different method?
5. What is the significance of the odd terms in the Taylor series for $\sin(x)$?
6. How do you find the Taylor series for a function that isn’t easily differentiable?
7. What are the applications of Taylor series in physics and engineering?
8. How does the approximation accuracy of the Taylor series change as you add more terms?

Tip: When working with Taylor series, always check the first few terms to see how well the series approximates the original function near the center.