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The cosine function, represented as \( \cos x \), is one of the fundamental trigonometric functions. It describes the horizontal coordinate of a point on the unit circle as you move around it. More intuitively, it tells us how far "right" or "left" a point lies on the circle as an angle \( x \) changes. The cosine function has the key property of being periodic with a period of \( 2\pi \), meaning every \( 2\pi \), its values repeat. Some important characteristics of \( \cos x \):

• It is an even function: \( \cos(-x) = \cos(x) \).
• It has a range of \(-1\) to \(1\), and the graph oscillates within this range.
• It is continuous and differentiable everywhere on the real line.

These properties make the cosine function well-suited for series expansions like the Taylor series.

When expanding functions such as \( \cos x \) using a Taylor series, understanding derivatives is crucial. The derivative of a function reflects its rate of change. For \( \cos x \), the sequence of derivatives forms a repeating pattern every four calculations:

• \( f(x) = \cos x \)
• \( f'(x) = -\sin x \)
• \( f''(x) = -\cos x \)
• \( f'''(x) = \sin x \)
• \( f^{(4)}(x) = \cos x \)

This cyclical pattern is a result of the periodic nature of trigonometric functions. Each derivative represents how the cosine function changes at different orders of approximation and directly affects the terms in the Taylor series.

Series expansion allows us to express functions as sums of simpler terms. The Taylor series is a powerful form of series expansion that expresses any function as an infinite sum based on its derivatives at a single point. The general formula is:\[T(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n\]For \( f(x) = \cos x \) centered at \( a = \frac{\pi}{2} \), it can be written as:\[T(x) = 0 \cdot (x-\frac{\pi}{2})^0 + (-1) \cdot \frac{(x-\frac{\pi}{2})^1}{1!} + 0 \cdot \frac{(x-\frac{\pi}{2})^2}{2!} + 1 \cdot \frac{(x-\frac{\pi}{2})^3}{3!} + \cdots\]From the pattern of derivatives, some terms vanish because their coefficients are zero. This leaves us with a pattern:\[T(x) = - (x-\frac{\pi}{2}) + \frac{(x-\frac{\pi}{2})^3}{3!} - \frac{(x-\frac{\pi}{2})^5}{5!} + \cdots\]This pattern highlights how only odd powers contribute to the expansion when centered at \( \frac{\pi}{2} \).

A Taylor series is centered at a specific point \( x = a \). This point is where the function and its derivatives are evaluated. By centering the series at this point, we create a polynomial that closely approximates the original function near \( a \). In this exercise, the Taylor series is centered at \( x = \frac{\pi}{2} \).Evaluating derivatives at this point:

• \( \cos\left(\frac{\pi}{2}\right) = 0 \)
• \( -\sin\left(\frac{\pi}{2}\right) = -1 \)
• \( -\cos\left(\frac{\pi}{2}\right) = 0 \)
• \( \sin\left(\frac{\pi}{2}\right) = 1 \)
• \( \cos\left(\frac{\pi}{2}\right) = 0 \)

This centering ensures that the Taylor series provides the best possible approximation around \( \frac{\pi}{2} \), reflecting how the cosine function behaves near this point.