Problem 87 - Algebra and Trigonometry

Question

(a) Show that there is a polynomial \(P(t)\) of degree 4 such that \(\cos 4 x=P(\cos x)\) (see Example 2\()\). (b) Show that there is a polynomial \(Q(t)\) of degree 5 such that \(\cos 5 x=Q(\cos x)\). NOTE In general, there is a polynomial \(P_{n}(t)\) of degree \(n\) such that \(\cos n x=P_{n}(\cos x) .\) These polynomials are called Tchebycheff polynomials, after the Russian mathematician P. L. Tchebycheff \((1821-1894)\) .

Step-by-Step Solution

Answer

Polynomials are \(P(t) = 8t^4 - 8t^2 + 1\) and \(Q(t) = 16t^5 - 20t^3 + 5t\).

Step 1: Understand the Problem

We need to express the cosine of multiple angles \(4x\) and \(5x\) in terms of a polynomial function of \(\cos x\) such as \(P(\cos x)\) and \(Q(\cos x)\). This aligns with Tchebycheff polynomials where \(\cos nx = P_n(\cos x)\).

Step 2: Decompose Cosine Using Angle Addition

For angles like \(nx\), the cosine function can be recursively expressed using the cosine addition formula: \(\cos((n-1)x + x) = \cos((n-1)x)\cos x - \sin((n-1)x)\sin x\).

Step 3: Recursion for \(\cos 4x\)

Use the recursion relation starting from a known basis: \(\cos 2x = 2\cos^2 x - 1\),\(\cos 3x = 4\cos^3 x - 3\cos x\).Apply the relation to express \(\cos 4x\) in terms of \(\cos x\) using previous expressions:\(\cos 4x = 2\cos(2x) \cos x - \sin^2(2x)\).Using \(\cos 2x\) and \(\sin^2 x = 1 - \cos^2 x\), solve for \(\cos 4x\).

Step 4: Calculate \(\cos 4x\)

Expand and simplify the expression:\(\cos 4x = 2(2\cos^2 x - 1)\cos x - (1 - \cos^2(2x))\)\(\cos 4x = 8\cos^4 x - 8\cos^2 x + 1\).Thus, the polynomial \(P(t) = 8t^4 - 8t^2 + 1\).

Step 5: Recursion for \(\cos 5x\)

Use the same angle addition formula:\(\cos 5x = \cos(4x + x) = \cos 4x \cos x - \sin 4x \sin x\).Substitute \(\cos 4x\) from Step 4 and simplify the expression to find \(\cos 5x\).

Step 6: Calculate \(\cos 5x\)

Using \(\cos 4x = 8\cos^4 x - 8\cos^2 x + 1\), replace and simplify to find \(\cos 5x\) in terms of \(\cos x\):\(\cos 5x = (8 \cos^4 x - 8 \cos^2 x + 1)\cos x - 5 (1 - \cos^2 x)(\cos x)\).Simplify to obtain \(\cos 5x = 16\cos^5 x - 20\cos^3 x + 5\cos x\), hence the polynomial \(Q(t) = 16t^5 - 20t^3 + 5t\).

Key Concepts

Polynomial Equations

Polynomial equations are expressions that involve variables raised to powers, typically found in mathematics for solving various problems. In our exercise, the aim was to express trigonometric expressions like \( \cos 4x \) and \( \cos 5x \) using polynomial forms, specifically through Tchebycheff polynomials. Polynomials can be used to approximate complex functions like trigonometric functions, enabling simpler calculations in some contexts. They consist of terms summed together, each being the product of a constant coefficient and a variable raised to a non-negative integer power. For example, the polynomial \( P(t) = 8t^4 - 8t^2 + 1 \) is composed of three terms: \( 8t^4 \), \(-8t^2 \), and \(1\), expressed as powers of \(t\). In trigonometry, presenting \( \cos nx \) as a polynomial in \( \cos x \) allows for easier manipulation and understanding of trigonometric identities. Each term in the polynomial represents a contribution from a specific angle or power, simplifying complex trigonometric operations.

Angle Addition

The angle addition formula is a fundamental tool in trigonometry that helps in expressing trigonometric functions of multiple angles. It states that the cosine of the sum of two angles can be broken down as:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

This identity is incredibly useful when dealing with expressions like \( \cos 4x \) and \( \cos 5x \) which can be expressed recursively. The process involves starting from a basic angle identity like \( \cos 2x = 2\cos^2 x - 1 \), then using the angle addition formula to derive further expressions. By applying the angle addition concept, you can break down the required expression into manageable computations, revealing patterns that can be converted into polynomial forms. Understanding angle addition is crucial for simplifying intricate trigonometric identities and connecting them with polynomial equations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. These identities are essential tools in solving trigonometric equations, simplifying expressions, and calculating angle values. For Tchebycheff polynomials, trigonometric identities such as the double angle formulas and angle sum formulas are key. Some basic trigonometric identities to remember include:

• \( \cos 2x = 2\cos^2 x - 1 \)
• \( \sin^2 x = 1 - \cos^2 x \)
• \( \cos 3x = 4\cos^3 x - 3\cos x \)

These identities help streamline the process of expressing \( \cos 4x \) and \( \cos 5x \) in polynomial terms. The transformation from trigonometric expressions to polynomials, using identities, illustrates a deep connection between different mathematical representations. Trigonometric identities thus serve as the bridge between angles and polynomials, enabling a unifying approach to different branches of mathematics.