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The problem asks to find the value of \( \cos\left(\frac{\pi}{3}\right) \) using the Taylor series expansion of \( \cos x \) to 5-digit accuracy, and then calculate the absolute, relative, and percentage errors. 

The Taylor series expansion for \( \cos x \) around \( x = 0 \) is given by:

\[
\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
\]

You are to evaluate \( \cos\left(\frac{\pi}{3}\right) \) using this series.

Let’s calculate the approximation for \( \cos\left(\frac{\pi}{3}\right) \) using the first few terms of the series to achieve 5-digit accuracy, and then proceed to compute the errors.

I'll first calculate the series approximation for \( \cos\left(\frac{\pi}{3}\right) \).The approximate value of \( \cos\left(\frac{\pi}{3}\right) \) using the Taylor series is:

\[
\cos\left(\frac{\pi}{3}\right) \approx 0.49996
\]

The exact value of \( \cos\left(\frac{\pi}{3}\right) \) is:

\[
\cos\left(\frac{\pi}{3}\right) = 0.50000
\]

### Errors:
- **Absolute error**: \( 3.5435 \times 10^{-5} \)
- **Relative error**: \( 7.0869 \times 10^{-5} \)
- **Percentage error**: \( 0.00709\% \)

Would you like more details or explanations about this process?

Here are 5 follow-up questions for further exploration:
1. What happens if you include higher-order terms in the series?
2. How does the Taylor series behave for values closer to zero?
3. What is the convergence rate of the Taylor series for cosine?
4. Can the Taylor series be used for other trigonometric functions like sine?
5. How does this method compare to using a calculator for finding cosine?

**Tip**: The Taylor series for cosine converges faster near \( x = 0 \), but for larger values of \( x \), more terms are required for higher accuracy.

Find the value of cos(x) at π/3 with 5-digit accuracy using the expansion cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + (x^8)/8!. Find absolute, relative and percentage error.

This problem demonstrates how to calculate cos(π/3) using the Taylor series expansion with 5-digit accuracy. Learn how to compute the value of cos(x) using the expansion formula, and how to find the absolute, relative, and percentage errors. Suitable for advanced high school or early college students studying trigonometry.

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