Problem 42 - Algebra and Trigonometry

Question

use words to describe the given formula. $$ \cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} $$

Step-by-Step Solution

Answer

The formula is expressing that the sum of the cosine of angle alpha and the cosine of angle beta equals two times the product of the cosine of half the sum of alpha and beta, and the cosine of half the difference between alpha and beta.

Step 1: Identify each element of the formula

The formula consists of four main parts: \( \cos \alpha \), \( \cos \beta \), \( \cos \frac{\alpha+\beta}{2} \), and \( \cos \frac{\alpha-\beta}{2} \). 'Cos' refers to the cosine function in trigonometry, and the Greek letters \( \alpha \) and \( \beta \) denote the angles. The sign + indicates the addition of \( \cos \alpha \) and \( \cos \beta \). The sign = denotes that the equation's left side equals the right side. The coefficient 2 multiples the result of \( \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \).

Step 2: Describe each element

\(\cos \alpha\) is the cosine of angle alpha, \(\cos \beta\) is the cosine of angle beta, \(2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}\) is two times the product of the cosine of half the sum of alpha and beta, and the cosine of half the difference between alpha and beta.

Step 3: Combine the descriptions

The sum of the cosine of angle alpha and the cosine of angle beta equals two times the product of the cosine of half the sum of alpha and beta, and the cosine of half the difference between alpha and beta.

Key Concepts

Cosine Function

The cosine function, commonly represented as \( \cos \), is a fundamental trigonometric function. It describes the relationship between the adjacent side and the hypotenuse in a right-angled triangle. Through this relationship, cosine becomes essential in various applications ranging from geometry to engineering.

• Definition: For an angle \( \theta \) in a right-angled triangle, the cosine is defined as \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• Range and Period: The values of cosine range from -1 to 1. It repeats every \(2\pi\) radians, showing its periodic nature.
• Graphical Representation: The graph of the cosine function is a wave that starts at 1 when the angle is 0, oscillating between -1 and 1.

Understanding the cosine function helps in visualizing and solving problems involving angles. The sum or differences of angles, as described in various identities, relies heavily on the properties of the cosine function.

Angle Addition Formulas

In trigonometry, angle addition formulas allow you to find the sine, cosine, or tangent of sum or difference of angles. These formulas are powerful tools for simplifying expressions and solving trigonometric equations.

• Cosine of Sum: For any two angles \( \alpha \) and \( \beta \), the cosine of their sum is given by \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).


• Cosine of Difference: Similarly, the cosine of their difference is \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \).
• Applications: These identities are not only useful in calculations but also serve as foundational blocks for deriving other trigonometric identities like sum-to-product.

Mastering these formulas allows for a deeper understanding of trigonometry as it relates to combined angles and their relationships.

Sum-to-Product Identities

The sum-to-product identities are trigonometric relationships that convert sums or differences of trigonometric functions into products. These identities simplify calculations and are particularly useful in integration and solving trigonometric equations.

• Key Identity: One of the main identities is \( \cos \alpha + \cos \beta = 2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \). This makes it possible to express the sum of cosines as a product of cosines, simplifying expressions significantly.


• Why It Works: The identities stem from the angle addition and subtraction formulas. By manipulating these formulas, the transformation between sums and products is achieved.
• Application: Sum-to-product identities prove useful in various branches of mathematics and physics, especially when dealing with waveforms and signals.

By using these identities, complex trigonometric expressions become more manageable, making them an essential tool in both theoretical and applied mathematics.