Problem 11 - Precalculus: Functions and Graphs

Question

Express as a product. $$\cos 5 x-\cos 3 x$$

Step-by-Step Solution

Answer

\( \cos 5x - \cos 3x = -2 \sin(4x) \sin(x) \).

Step 1: Identify the Formula

To express \( \cos 5x - \cos 3x \) as a product, we need to use a trigonometric identity specifically for the difference of cosines: \[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]. Here, \( A = 5x \) and \( B = 3x \).

Step 2: Apply the Identity

Using the formula \( A = 5x \) and \( B = 3x \), we first calculate \( \frac{A+B}{2} = \frac{5x + 3x}{2} = 4x \) and \( \frac{A-B}{2} = \frac{5x - 3x}{2} = x \).

Step 3: Express as a Product

Now substitute the values back into the identity: \[ \cos 5x - \cos 3x = -2 \sin(4x) \sin(x) \].

Step 4: Conclusion

The expression \( \cos 5x - \cos 3x \) is now expressed as a product of two sine functions: \[ -2 \sin(4x) \sin(x) \].

Key Concepts

Difference of Cosines

When dealing with trigonometric expressions, it's crucial to understand how different identities transform them. One such identity is the 'difference of cosines.' This particular identity allows us to express a difference between two cosine terms as a product of sine terms. For the expression \( \cos A - \cos B \), the identity is:

• \( \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \)

This formula transforms the trigonometric difference into a product, making it easier to simplify or analyze the expression further. The essence of this transformation lies in leveraging symmetry and periodic properties of cosine and sine functions. This can help in various calculus and physics problems where simplification is necessary.

Product-to-Sum Identities

Product-to-sum identities are a powerful tool in trigonometry. They go hand in hand with their counterparts, the sum-to-product identities. These are used to convert products of trigonometric functions into sums or differences. In the context of the 'difference of cosines,' employing this identity offers a neat way to handle multi-angle expressions. For example, expressing terms like \( \cos 5x - \cos 3x \) as a product simplifies the expression. Using these identities appropriately can:

• Simplify integration and differentiation of trigonometric functions.
• Facilitate solving trigonometric equations or inequalities.
• Allow easier manipulation within signal processing or oscillation problems.

Knowing when and how to switch between these forms is key to mastering trigonometry applications.

Trigonometry

Trigonometry is a branch of mathematics focused on the relationships between the angles and sides of triangles. It's divided into a few key functions: sine, cosine, and tangent, each describing a specific ratio relevant to the angle of interest. The study of these functions extends beyond simple triangle analysis. It involves understanding wave functions and circular motion, which is why identities like 'difference of cosines' and 'product-to-sum' are pivotal. Also, trigonometry is essential in various fields such as:

• Physics, particularly in wave motion, optics, and electromagnetism.
• Engineering, for handling oscillations and signal processing.
• Computer graphics, for rendering and animation calculation.

Mastering trigonometric identities and their transformations opens a myriad of analytical and practical applications.