Problem 17 - Advanced Engineering Mathematics

Question

\(\mathrm{U} \operatorname{sing} \cos ^{2} \theta=\frac{1}{2}(\cos 2 \theta+1)\) we have $$\begin{aligned} P_{2}(\cos \theta) &=\frac{1}{2}\left(3 \cos ^{2} \theta-1\right)=\frac{3}{2} \cos ^{2} \theta-\frac{1}{2} \\ &=\frac{3}{4}(\cos 2 \theta+1)-\frac{1}{2}=\frac{3}{4} \cos 2 \theta+\frac{1}{4}=\frac{1}{4}(3 \cos 2 \theta+1), \end{aligned}$$

Step-by-Step Solution

Answer

We simplify \(P_2(\cos \theta)\) to \(\frac{1}{4}(3\cos 2\theta + 1)\).

Step 1: Understanding the Problem

We are given an expression for \(\cos^2 \theta\) and a polynomial \(P_2(\cos \theta)\). The task is to verify the transformation of \(P_2(\cos \theta)\) using the expression of \(\cos^2 \theta\).

Step 2: Expand \(P_2(\cos \theta)\) using \(\cos^2 \theta\)

Start by replacing \(\cos^2 \theta\) with \(\frac{1}{2}(\cos 2 \theta + 1)\) in the expression for \(P_2(\cos \theta)\): \[P_2(\cos \theta) = \frac{3}{2}\left(\frac{1}{2}(\cos 2 \theta + 1)\right) - \frac{1}{2}\]

Step 3: Simplify the Expression

Simplify the expression obtained in Step 2:\[P_2(\cos \theta) = \frac{3}{4}(\cos 2 \theta + 1) - \frac{1}{2}\]This becomes:\[P_2(\cos \theta) = \frac{3}{4}\cos 2 \theta + \frac{3}{4} - \frac{2}{4}\]

Step 4: Final Simplification

Continue simplifying the expression:\[P_2(\cos \theta) = \frac{3}{4}\cos 2 \theta + \frac{1}{4}\]And thus,\[P_2(\cos \theta) = \frac{1}{4}(3\cos 2 \theta + 1)\]

Step 5: Verification

The transformation and simplification lead to the desired expression:\(P_2(\cos \theta) = \frac{1}{4}(3 \cos 2 \theta + 1)\), verifying the initial condition given in the problem.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics. They allow us to transform and simplify expressions involving trigonometric functions. In this exercise, we're particularly interested in the identity for \(\cos^2 \theta\). The identity \(\cos^2 \theta = \frac{1}{2} ( \cos 2\theta + 1)\) is one of the double-angle identities.

These identities are valuable because they express trig functions in terms of angles twice as large or in forms that make calculations easier. Knowing these identities helps when working with polynomials involving trigonometric functions, allowing transformations and simplifications that reveal more details or make functions easier to integrate or differentiate.

• Example: \(\cos^2 \theta\) can be written as \(\frac{1}{2}(\cos 2\theta + 1)\).
• Purpose: Simplifies expressions to make further calculations manageable.

By understanding how these identities work, you can manipulate expressions to solve complex mathematical challenges more intuitively.

Polynomial Transformation

Polynomial transformation refers to altering a polynomial expression's form while retaining its core properties. This concept is evident in the task involving the Legendre polynomial \(P_2(x)\). Given as \(\frac{1}{2}(3x^2 - 1)\), this polynomial is transformed using a trigonometric identity.

Here, the identity \(\cos^2 \theta = \frac{1}{2}(\cos 2\theta + 1)\) allows us to express \(x\) in terms of \(\cos 2\theta\), effectively transforming the polynomial into an expression that's easier to work with.

• Initial Form: \(P_2(\cos \theta) = \frac{1}{2}(3\cos^2 \theta - 1)\)
• Transformed Form: Uses the identity to become \(\frac{1}{4}(3\cos 2\theta + 1)\).

This transformation doesn't change the polynomial's roots or basic behavior but instead refines its expression. This is crucial in solving integrals and verifying properties across different mathematical tasks.

Mathematical Verification

Mathematical verification ensures that our transformations and calculations result in correct outcomes. In trigonometric and polynomial exercises, verifying each step is crucial to confirm our results are accurate and consistent with known identities and formulas.

To verify the transformation in this exercise, we start by substituting the trigonometric identity into the polynomial and simplifying along each step. By doing so, we ensure that any errors can be traced and corrected efficiently.

• Initial Verification: Replace \(\cos^2 \theta\) with its identity form.
• Simplification: Rework the expression to show that it matches the given form.
• Final Check: Confirm that the achieved expression \(\frac{1}{4}(3\cos 2\theta + 1)\) aligns with the expected result.

Verification is vital as it provides the confidence that transformations are performed correctly, ensuring that equations are not only simplified but also equivalent to the original form. This builds a strong foundation for tackling more complex mathematical challenges.