To find the Taylor series expansion of the function $\sin(x)$ centered at $x = \frac{\pi}{4}$, we will express $\sin(x)$ as a Taylor series around $x = \frac{\pi}{4}$.

Taylor Series Formula

The Taylor series of a function $f(x)$ centered at $x = a$ is given by:

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

In this case, $f(x) = \sin(x)$ and $a = \frac{\pi}{4}$.

Step-by-Step Calculation

1. Compute the derivatives of $\sin(x)$:

f(x) &= \sin(x), \\ f'(x) &= \cos(x), \\ f''(x) &= -\sin(x), \\ f'''(x) &= -\cos(x), \\ f^{(4)}(x) &= \sin(x), \quad \text{(and repeats in cycles)} \end{align*}$$ 2. **Evaluate the derivatives at $$x = \frac{\pi}{4}$$:** $$\begin{align*} f\left(\frac{\pi}{4}\right) &= \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \\ f'\left(\frac{\pi}{4}\right) &= \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \\ f''\left(\frac{\pi}{4}\right) &= -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \\ f'''\left(\frac{\pi}{4}\right) &= -\cos\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}, \\ f^{(4)}\left(\frac{\pi}{4}\right) &= \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}. \end{align*}$$ 3. **Construct the Taylor series:** Substitute these values into the Taylor series formula: $$\sin(x) \approx \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \cdot (x - \frac{\pi}{4}) - \frac{\sqrt{2}}{4} \cdot (x - \frac{\pi}{4})^2 - \frac{\sqrt{2}}{12} \cdot (x - \frac{\pi}{4})^3 + \cdots$$ ### Simplified Taylor Series for $$\sin(x)$$ around $$x = \frac{\pi}{4}$$: $$\sin(x) \approx \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} (x - \frac{\pi}{4}) - \frac{\sqrt{2}}{4} (x - \frac{\pi}{4})^2 - \frac{\sqrt{2}}{12} (x - \frac{\pi}{4})^3 + \cdots$$ This gives the Taylor expansion up to the third degree. You can continue the series by calculating higher-order derivatives and including more terms as needed. Would you like to see more details or have any questions about this process? ### Additional Related Questions: 1. How do you derive the Taylor series for other trigonometric functions like $$\cos(x)$$ around different points? 2. What is the radius of convergence for the Taylor series of $$\sin(x)$$? 3. How can you use Taylor series to approximate $$\sin(x)$$ for small values of $$x$$? 4. What is the significance of choosing a different center point for a Taylor series expansion? 5. How do you determine the error or remainder of a Taylor series approximation? #### Tip: When approximating a function using a Taylor series, the more terms you include, the closer the approximation will be to the original function, especially near the center point $$x = a$$.