Problem 8 - Precalculus Essentials

Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\cos \frac{5 x}{2} \sin \frac{x}{2}$$

Step-by-Step Solution

Answer

The expression \(\cos \frac{5 x}{2} \sin \frac{x}{2}\) can be expressed as a sum by using the product-to-sum identity, resulting in \(\frac{1}{2} [\sin (3x) + \sin (2x)]\).

Step 1: Identify the values of a and b

In the given expression \(\cos \frac{5 x}{2} \sin \frac{x}{2}\), identify the values of a and b from the product-to-sum identity. Here, \(a = \frac{5x}{2}\) and \(b = \frac{x}{2}\).

Step 2: Substitute the values of a and b into the identity

Into the product-to-sum identity \(\sin a \cos b = \frac{1}{2} [\sin (a + b) + \sin (a - b)]\), substitute \(a = \frac{5x}{2}\) and \(b = \frac{x}{2}\). This gives us \(\cos \frac{5 x}{2} \sin \frac{x}{2} = \frac{1}{2} [\sin (\frac{5x}{2} + \frac{x}{2}) + \sin (\frac{5x}{2} - \frac{x}{2})]\).

Step 3: Simplify the expression

Simplify the expression in the bracket. This results in \(\cos \frac{5 x}{2} \sin \frac{x}{2} = \frac{1}{2} [\sin (3x) + \sin (2x)]\).

Key Concepts

Product-to-Sum Identities

Product-to-sum identities are a set of trigonometric formulas that allow us to express the product of trigonometric functions as a sum or difference. This can simplify complex trigonometric expressions, making them easier to work with.
For example, in the given exercise, we have the expression \( \cos \frac{5x}{2} \sin \frac{x}{2} \). This expression seems complex initially, but by using a product-to-sum identity, it can be transformed.
The identity used here is:

• \( \sin a \cos b = \frac{1}{2} [\sin (a + b) + \sin (a - b)] \)

By substituting the values \( a = \frac{5x}{2} \) and \( b = \frac{x}{2} \) into the identity, the given product is expressed as a sum: \( \frac{1}{2} [\sin (3x) + \sin (2x)] \).
This simplifies calculations and provides a different perspective on solving trigonometric problems.

Trigonometric Functions

Trigonometric functions form the backbone of trigonometry, and they are fundamental in expressing relations in a triangle. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function represents a specific ratio of sides of a right-angle triangle.
In our exercise, the functions involved are cosine and sine.

• Cosine (\(\cos\)): In a right triangle, cosine represents the ratio of the length of the adjacent side to the hypotenuse.
• Sine (\(\sin\)): This function represents the ratio of the length of the opposite side to the hypotenuse.

These functions are not only essential in geometry but also play a critical role in various areas such as physics and engineering, especially when analyzing waveforms and oscillations.

Trigonometry

Trigonometry is a branch of mathematics dealing with the relationships between the angles and sides of triangles, particularly right-angled triangles. It is an essential tool in the field of mathematics, aiding in the understanding of angles and distances.
Using trigonometry, one can find missing angles and sides of triangles, model periodic phenomena, and even solve equations in fields like physics, engineering, and astronomy.
In the realm of pure math, identities like the product-to-sum identities are particularly useful. They enable mathematicians and scientists to simplify expressions and solve problems more efficiently.

By mastering trigonometry, one can unlock a vast array of techniques and applications that are valuable across many scientific disciplines, making it a crucial area of study for students and professionals alike.