Problem 50 - Algebra and Trigonometry

Question

47–52 Write the sum as a product. $$\cos 9 x+\cos 2 x$$

Step-by-Step Solution

Answer

The sum \( \cos 9x + \cos 2x \) can be written as the product \( 2 \cos \left(\frac{11x}{2}\right) \cos \left(\frac{7x}{2}\right) \).

Step 1: Recall the Trigonometric Identity

We will use the sum-to-product identity for cosine functions: \ \( \cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right) \). This identity allows us to express the sum of two cosines as a product of two cosines.

Step 2: Identify the Values of A and B

Given the expression \( \cos 9x + \cos 2x \), identify \( A = 9x \) and \( B = 2x \).

Step 3: Calculate (A+B)/2 and (A-B)/2

Now, calculate \( \frac{A + B}{2} = \frac{9x + 2x}{2} = \frac{11x}{2} \) and \( \frac{A - B}{2} = \frac{9x - 2x}{2} = \frac{7x}{2} \).

Step 4: Apply the Identity

Substitute \( A \) and \( B \) into the sum-to-product formula to rewrite the expression: \ \( \cos 9x + \cos 2x = 2 \cos \left(\frac{11x}{2}\right) \cos \left(\frac{7x}{2}\right) \).

Step 5: Write the Final Product

The given sum \( \cos 9x + \cos 2x \) is expressed as a product: \ \( 2 \cos \left(\frac{11x}{2}\right) \cos \left(\frac{7x}{2}\right) \).

Key Concepts

Sum-to-Product Identities

When dealing with trigonometric expressions, sum-to-product identities are valuable tools. They help convert the sum of trigonometric functions into a product, making them simpler to manipulate or compute. For example, the identity for the sum of two cosine functions is:\[ \cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right) \]This particular identity is useful because it transforms two separate cosine terms into a single expression involving a product. This form can often simplify integration and other calculations in trigonometry. Sum-to-product identities often appear in problems where expression manipulation is required to align with further operations. Always pay attention to the formulas for different trigonometric functions, such as sine and cosine, as they can greatly simplify complex expressions.

Cosine Function

The cosine function is one of the basic trigonometric functions, denoted as \( \cos \). It describes the adjacent side over the hypotenuse in a right triangle or the horizontal coordinate in the unit circle representation.

• On the unit circle, \( \cos(\theta) \) is the x-coordinate at angle \( \theta \).
• The function is periodic with a period of \( 2\pi \), which means \( \cos(\theta + 2\pi) = \cos(\theta) \).
• It is even, meaning \( \cos(-\theta) = \cos(\theta) \).

In expressions, the cosine function often appears in identities and transformations. Knowing these properties helps when simplifying trigonometric equations or using identities such as the sum-to-product identity.

Product of Trigonometric Functions

Trigonometric identities often transform sums or differences into products. This technique leverages the unit circle and wave properties of trigonometric functions to simplify expressions. In the context of our problem, using the sum-to-product identity, we can rewrite sums as products, which helps in various calculations such as integration.

• The transformation process often makes complex expressions easier to handle.
• Products of trigonometric functions can reveal important properties, such as amplitude changes or phase shifts.
• This manipulation is common in solving physics problems where wave interference or signal processing is concerned.

Understanding how to transform sums into products prepares you for more advanced topics in calculus and beyond, where these skills are crucial.