To find the Taylor series expansion of the function $\cos(x)$ about $x = \frac{\pi}{4}$, we need to determine the function's derivatives at $x = \frac{\pi}{4}$ and then construct the series.

The Taylor series expansion of $\cos(x)$ about $x = \frac{\pi}{4}$ is given by:

$\cos(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}\left(\frac{\pi}{4}\right)}{n!} \left(x - \frac{\pi}{4}\right)^n$

where $f^{(n)}\left(\frac{\pi}{4}\right)$ represents the $n$-th derivative of $\cos(x)$ evaluated at $x = \frac{\pi}{4}$.

Step 1: Compute the derivatives of $\cos(x)$

Let's compute the first few derivatives of $\cos(x)$:

$f(x) = \cos(x)$

$f'(x) = -\sin(x)$

$f''(x) = -\cos(x)$

$f'''(x) = \sin(x)$

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

The pattern repeats every four derivatives.

Step 2: Evaluate the derivatives at $x = \frac{\pi}{4}$

$f\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

$f'\left(\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

$f''\left(\frac{\pi}{4}\right) = -\cos\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$

$f'''\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Step 3: Write the Taylor series

Substituting these into the Taylor series formula, we have:

$\cos(x) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \cdot \left(x - \frac{\pi}{4}\right) - \frac{\sqrt{2}}{4} \cdot \left(x - \frac{\pi}{4}\right)^2 + \frac{\sqrt{2}}{12} \cdot \left(x - \frac{\pi}{4}\right)^3 + \cdots$

Final Taylor Series Expansion

The Taylor series expansion of $\cos(x)$ about $x = \frac{\pi}{4}$ is:

$\cos(x) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}\left(x - \frac{\pi}{4}\right) - \frac{\sqrt{2}}{4}\left(x - \frac{\pi}{4}\right)^2 + \frac{\sqrt{2}}{12}\left(x - \frac{\pi}{4}\right)^3 + \cdots$

This is the Taylor series expansion of $\cos(x)$ centered at $x = \frac{\pi}{4}$.

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

Related Questions:

1. How do you find the radius of convergence for the Taylor series?
2. Can you find the Taylor series of $\cos(x)$ about $x = 0$ (Maclaurin series)?
3. What are the applications of Taylor series in physics?
4. How can you approximate $\cos(x)$ near $x = \frac{\pi}{4}$ using the first few terms of the series?
5. How does the Taylor series change if the function was $\sin(x)$ instead of $\cos(x)$?

Tip:

To improve accuracy, the more terms you include in the Taylor series, the better the approximation of the function near the point of expansion.