Problem 6 - Advanced Engineering Mathematics

Question

Jse the Maclaurin series for \(\sin x\) and \(\cos x\) along with long livision to find the first three nonzero terms of a power series n \(x\) for the function \(f(x)=\frac{\sin x}{\cos x}\).

Step-by-Step Solution

Answer

The first three nonzero terms are \( x + \frac{x^3}{3} + \frac{x^5}{5} \).

Step 1: Write the Maclaurin Series for sin(x) and cos(x)

The Maclaurin series for \( \sin x \) is \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \), and for \( \cos x \) is \( \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \). These series will help to express both sine and cosine as power series expansions.

Step 2: Set Up the Division

We need to find the series for \( \frac{\sin x}{\cos x} \). This requires long division of the series for \( \sin x \) by the series for \( \cos x \). Start with the leading terms: the numerator \( \sin x \approx x \) and the denominator \( \cos x \approx 1 \).

Step 3: Perform Long Division

Perform the polynomial long division using the series: divide \( x \) by \( 1 \), getting \( x \) as the first term. Continue by subtracting \( x \) times the series of \( \cos x \), focusing on the next terms: divide \( -\frac{x^3}{6} + \cdots \) by \( 1 \), yielding \( +\frac{x^3}{3} \) as the second term. Continue similarly to find \( +\frac{x^5}{5} \) as the third term.

Step 4: Collect the Power Series Terms

After the long division, gather the terms. The function \( f(x) = \frac{\sin x}{\cos x} \) is approximated by the power series: \( x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots \). This represents the first three nonzero terms of the power series.

Key Concepts

Power Series

Understanding power series is crucial when working with functions like sine and cosine. A power series is an infinite sum of terms that looks like this: \( a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots \), where \(x\) is a variable and the \(a_n\)'s are constants, often expressed in terms of derivatives evaluated at a specific point. For the Maclaurin series, this point is zero, which simplifies calculations considerably.
Maclaurin series are specific types of power series used to represent functions:

• For \( \sin x \), it's given by: \( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \).
• For \( \cos x \), it's given by: \( \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \).

These series demonstrate how trigonometric functions can be expanded into an infinite series of polynomial terms. Using power series, we can approximate complex functions with just a few terms, which simplifies many mathematical tasks.
In this exercise, using the first few terms from the Maclaurin series can provide a clear approximation of the function \( \frac{\sin x}{\cos x} \). This introduces us to the world of power series and their power in simplifying expressions.

Long Division

Polynomial long division is a technique that can extend our understanding of division from numbers to polynomials. When we divide a polynomial by another polynomial, we can find the quotient, remainder, or in some cases, a simplified expression.
In this exercise, the goal is to divide the Maclaurin series of \( \sin x \) by \( \cos x \) to derive the power series for \( \frac{\sin x}{\cos x} \). Start by considering the leading terms: if we approximate \( \sin x \approx x \) and \( \cos x \approx 1 \), the division seems simple initially.

• The first step should approximate \( \frac{x}{1} = x \).
• Subtract the approximation of \( x \) times \( \cos x \) from \( \sin x \) to find the next term.

Continue this process through the series, treating it like repeated division with remainders until the desired number of terms is reached, achieving a series such as \( x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots \).
This long division process is essential, as it allows us to tailor the power series to specific functions, giving us approximate models that are easier to manipulate.

Trigonometric Functions

Trigonometric functions like sine and cosine are foundational in mathematics. They relate the angles of triangles to lengths, but they also have crucial applications in calculus and series.
The essence of trigonometry can be captured in power series, which simplifies working with angles and facilitates analysis. The Maclaurin series for sine and cosine reveal how these functions can be expressed in terms of polynomials straight from calculus:

• \( \sin x \) becomes \( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \).
• \( \cos x \) becomes \( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \).

When these series are employed together, as seen in this exercise, they aid in deriving series for other trigonometric identities. For example, \( \frac{\sin x}{\cos x} \) is the identity for \( \tan x \), which can similarly be expressed as a power series using long division of \( \sin x \) and \( \cos x \). Understanding these connections deepens our grasp of how trigonometric functions behave and makes complex calculations more approachable.