Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the sum-to-product formulas to write the sum or difference as a product. $$\cos 6 x+\cos 2 x$$

Step-by-Step Solution

Answer

The sum \( \cos 6x + \cos 2x \) can be rewritten as a product using the sum-to-product formulas as \( 2 \cos 4x \cos 2x \)

Step 1: Identify the cosine functions

In the expression \( \cos 6x + \cos 2x \), we have two cosine functions, \( \cos 6x \) and \( \cos 2x \). These acts as our \( A \) and \( B \) values in the sum-to-product formula.

Step 2: Apply the sum-to-product formula

Substituting the values into the sum-to-product formula, \[ \cos 6x + \cos 2x = 2 \cos \frac{1}{2}(6x + 2x) \cos \frac{1}{2}(6x - 2x) \]

Step 3: Simplify the equation

Upon simplifying, we get\[ = 2 \cos 4x \cos 2x \]

Key Concepts

Cosine Functions

Cosine functions are an essential part of trigonometry. They are periodic functions, meaning they repeat their values in regular intervals. This property is defined by the cosine wave that oscillates between -1 and 1. The cosine function is usually denoted as \( \cos(x) \), where \( x \) can be any real number or angle in radians. The fundamental period of the cosine function is \( 2\pi \).

• Symmetry: Cosine functions are even functions, meaning \( \cos(-x) = \cos(x) \).
• Zero Points: Cosine equals zero at odd multiples of \( \frac{\pi}{2} \).
• Maximum and Minimum Values: The maximum value is 1 (at multiples of \( 2\pi \)), and the minimum value is -1 (at odd multiples of \( \pi \)).

Understanding cosine functions is crucial, especially when transforming or manipulating them using various trigonometric identities, such as sum-to-product formulas.

Trigonometric Identities

Trigonometric identities are mathematical equations that express fundamental trig relationships. They help us simplify complex trig expressions and solve trig equations. One of the essential sets of these identities is the sum-to-product identities, which convert sums or differences of trigonometric functions into products.

For example, the sum-to-product formula for cosine functions is:\[ \cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]This formula is useful when dealing with the sum of two cosines, as it can simplify expressions by transforming them into a product of two trigonometric expressions. This is particularly helpful in analysis and problem-solving, enabling easier simplification and calculation. Utilizing identities efficiently requires practice and familiarity with the different formulas.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. In the context of trigonometry, these expressions often involve trigonometric functions that can be modified using various algebraic techniques.

• Simplification: Simplification involves reducing expressions to their simplest form using factoring, distributing, or applying identities, like the sum-to-product identities.
• Substitution: This involves replacing one part of the expression with another, equivalent expression. This is essential when applying trigonometric identities.

In the given exercise, simplifying expressions using sum-to-product formulas involves recognizing patterns within the algebraic structure of the trigonometric functions. Mastering both algebraic manipulations and understanding of trigonometric identities is key to solving such problems effectively.