approximate (sin(x))/x using the taylor series of sin(x) of degree 3 at x=0

To approximate the function \(\frac{\sin(x)}{x}\) using the Taylor series of \(\sin(x)\) of degree 3 centered at \(x = 0\), we can follow these steps:

### Step 1: Write the Taylor series for \(\sin(x)\) centered at \(x = 0\)
The Taylor series for \(\sin(x)\) around \(x = 0\) is given by:

\[
\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots
\]

If we only consider up to the third-degree term:

\[
\sin(x) \approx x - \frac{x^3}{6}
\]

### Step 2: Divide by \(x\) to approximate \(\frac{\sin(x)}{x}\)
Now, we approximate \(\frac{\sin(x)}{x}\) by dividing the Taylor series of \(\sin(x)\) by \(x\):

\[
\frac{\sin(x)}{x} \approx \frac{x - \frac{x^3}{6}}{x}
\]

Simplifying this:

\[
\frac{\sin(x)}{x} \approx 1 - \frac{x^2}{6}
\]

### Final Approximation
Thus, the approximation of \(\frac{\sin(x)}{x}\) using the Taylor series of \(\sin(x)\) of degree 3 centered at \(x = 0\) is:

\[
\frac{\sin(x)}{x} \approx 1 - \frac{x^2}{6}
\]

This approximation is valid for values of \(x\) close to 0.

Would you like any further details or have any questions?

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Here are 5 related questions you might find interesting:

1. How does the approximation change if we use a higher-degree Taylor series for \(\sin(x)\)?
2. What is the actual value of \(\frac{\sin(x)}{x}\) at \(x = 0.1\), and how does it compare to the approximation?
3. Can you derive a general formula for the Taylor series of \(\frac{\sin(x)}{x}\) at \(x = 0\)?
4. What happens to the approximation as \(x\) increases away from 0?
5. How can you use the remainder term to estimate the error in this approximation?

**Tip:** The approximation \(\frac{\sin(x)}{x} \approx 1 - \frac{x^2}{6}\) is particularly useful for small values of \(x\) in fields like physics and engineering, where such expressions often appear.

Explore how to approximate the function sin(x)/x using the Taylor series of sin(x) of degree 3 at x=0. Understand the steps involved and the accuracy of the approximation. Suitable for undergraduate students and those interested in mathematical approximation methods.

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