91Ó°ÊÓ

The cosine function, denoted as \(\cos x\), is a fundamental trigonometric function. It's periodic, with a cycle repeating every \(2\pi\) radians. The function's graph is a smooth wave that oscillates between -1 and 1. This function is essential in various branches of mathematics and physics, representing the x-coordinate of a unit circle at a given angle. Because it is differentiable everywhere and smooth, it pairs well with Taylor series expansion.When expanding \(\cos x\) into a Taylor series, the function provides a perfect model to demonstrate how infinite polynomials can approximate complex functions accurately. Knowing that \(\cos(\pi/3) = 1/2\) is beneficial because it serves as the initial term in our series expansion.

Derivatives are the backbone of Taylor series expansions. They demonstrate how a function changes at any point. For the cosine function \(f(x) = \cos x\), its derivatives alternate cyclically:

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

This pattern repeats indefinitely. Understanding this cycle is crucial because each derivative, evaluated at a specific point, contributes to the respective term of the Taylor series. In context, this cyclical nature means that generating terms beyond the first few becomes predictable and systematic.

To build a Taylor series for any function, you must evaluate its derivatives at a specific point—this point is often denoted as \(c\). In our case, \(c = \pi/3\). For each derivative of \(\cos x\), we substitute \(\pi/3\) and calculate the result:

• \(f(c) = \cos(\pi/3) = 1/2\)
• \(f'(c) = -\sin(\pi/3) = -\sqrt{3}/2\)
• \(f''(c) = -\cos(\pi/3) = -1/2\)
• \(f'''(c) = \sin(\pi/3) = \sqrt{3}/2\)

This evaluation is necessary because all these values define the coefficients in the series expansion terms. It lays down the foundation for constructing the Taylor series.

Constructing a Taylor series involves assembling an infinite series of terms based on:

• The initial function value at the point
• The values of the derivatives at that point

The general formula for a Taylor series expanded around a point \(c\) for a function \(f(x)\) is given by the sum:\[\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x-c)^n\]In our exercise, because each derivative of \(\cos x\) evaluated at \(\pi/3\) defines a coefficient, the series starts with \(\frac{1}{2}\) and includes alternating terms like \(-\frac{\sqrt{3}}{2}(x-\frac{\pi}{3})\) up to infinite degrees of \(x-c\).The constructive process involves plugging these coefficients into the Taylor series formula, culminating in an approximated polynomial expression for \(\cos x\) around the point \(c = \pi/3\). This expression becomes more accurate as more terms are added.