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The Taylor Series is a powerful tool used to approximate functions with polynomials. This series expands a function into an infinite sum of terms calculated from the function's derivatives at a single point. It provides an approximation that becomes more accurate as more terms are included.
For a function \( f(x) \) that is infinitely differentiable at a point \( a \), the Taylor series is expressed as:

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

In practice, we often use only the first few terms of the series to approximate functions like the cosine and sine. This is because computing infinite terms is not feasible. The number of terms we choose affects the accuracy of the approximation.

The notion of the Error Bound is crucial when working with Taylor series approximations. It provides a quantifiable measure of how accurate the polynomial approximation is compared to the original function.
For an \( n \)-degree Taylor polynomial, the error or remainder term \( R_n(x) \) represents the difference between the true function value and the polynomial approximation. The error bound helps us understand how large this remainder can be, ensuring we know the worst-case scenario in terms of deviation from the true function.
In our problem, for \( \cos x \) approximated by \( 1 - \frac{x^2}{2} \), the error bound was estimated to be \( \frac{1}{48} \), while for \( \sin x \) approximated by \( x - \frac{x^3}{6} \), it was \( \frac{1}{384} \). These calculations assure us within a specific bound how closely our polynomial matches the actual function for a given \( x \).

Approximating the cosine function with a Taylor series allows us to simplify its computation to a polynomial. The basic approximation used is \( \cos x \approx 1 - \frac{x^2}{2} \). This is a 2nd-degree polynomial derived from the Taylor series expansion of \( \cos x \).
Further approximations can be made by adding more terms, which increases accuracy. The truncated series \( 1 - \frac{x^2}{2} \) is often sufficient for small values of \( x \), especially since you're given \( |x| < \frac{1}{2} \), which ensures that terms beyond the second have a limited impact.
In calculating the error, our task is to find how far off this approximation might be from the actual \( \cos x \). By checking the remainder term or error, we find it's bounded by \( \frac{1}{48} \) in this specific exercise scenario.

When approximating the sine function, a similar process is followed using its Taylor series expansion. For \( \sin x \), an effective polynomial approximation is \( x - \frac{x^3}{6} \), which is a 3rd-degree polynomial.
This polynomial provides a simple way to compute the sine of small angles when high precision isn't necessary. Again, due to \( |x| < \frac{1}{2} \), the approximation is quite good, as higher order terms (like \( x^5 \) and beyond) are small.
Checking the error bound allows us to conclude that the maximum deviation from the true sine function will not exceed \( \frac{1}{384} \). Understanding this builds confidence in using polynomial approximations in practical applications where exact calculations are cumbersome or unnecessary.