Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the product as a sum or difference. \(2 \cos 2 \theta \cos 5 \theta\)

Step-by-Step Solution

Answer

The product can be written as a sum: \(\cos(3 \theta) + \cos(7 \theta)\)

Step 1 Apply product-to-sum identity

Apply the cosine product-to-sum identity to \(2 \cos 2 \theta \cos 5 \theta\), such that \(\alpha = 2 \theta\) and \(\beta = 5 \theta\). This means rewriting the expression as \(2 * \frac{1}{2}[\cos(2 \theta -5 \theta) + \cos(2 \theta + 5 \theta)]\)

Step 2 Simplify the expression

Simplify the expressions within the brackets to get \(\cos(-3 \theta) + \cos(7 \theta)\)

Step 3 Accounting for cosine's even property

Since cosine is an even function, meaning \(\cos(-x) = \cos(x)\), the expression \(\cos(-3 \theta)\) can be rewritten as \(\cos(3 \theta)\).

Key Concepts

Product-to-Sum Identities

Trigonometric identities simplify complex expressions into more manageable forms. One important identity is the product-to-sum identity. This identity transforms a product of trigonometric functions into a sum or difference of trigonometric terms.
In this case, the product-to-sum identity for cosine is:

• \(\cos \alpha \cos \beta = \frac{1}{2} [\cos (\alpha - \beta) + \cos (\alpha + \beta)]\)

Application: If you have an expression like \(2 \cos 2\theta \cos 5\theta\), you can first factor out the 2 and then use the product-to-sum identity to rewrite it. This results in:

• \(2 * \frac{1}{2} [\cos (2\theta - 5\theta) + \cos (2\theta + 5\theta)]\)

This identity is incredibly useful for integrals and simplifying algebraic expressions involving trigonometric functions.

Cosine Function

The cosine function is a fundamental part of trigonometry. It relates the angle of a right triangle to the lengths of the adjacent side and the hypotenuse.

Key Properties of Cosine:

• Range: The values of the cosine function range from -1 to 1.
• Periodicity: Cosine has a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
• Symmetry: It is an even function, which will be discussed further below.

The cosine function forms the backbone of various trigonometric identities, allowing complex expressions to be manipulated into simpler forms for analysis and calculus.

Even Function Property

An even function is symmetric around the y-axis. This means it produces the same result for both positive and negative versions of the same input.

Cosine's Even Property:

• For cosine, this property is expressed as: \(\cos(-x) = \cos(x)\).

In our problem, one of the expressions was \(\cos(-3\theta)\). Because cosine is even, we can rewrite this as \(\cos(3\theta)\).

Importance: This property simplifies problem-solving and allows mathematicians to manage trigonometric expressions more efficiently. Understanding and applying these symmetric properties help in reducing computation and enhancing clarity.