Problem 64 - Algebra and Trigonometry

Question

\(61-66\) Write the sum as a product. $$ \cos 9 x+\cos 2 x $$

Step-by-Step Solution

Answer

\(2 \cos \left( \frac{11x}{2} \right) \cos \left( \frac{7x}{2} \right)\)

Step 1: Identify the trigonometric identity

Recognize that the expression \( \cos 9x + \cos 2x \) can be written as a product using the sum-to-product identities: \[ \cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right). \] Identifying \( A = 9x \) and \( B = 2x \) is crucial for proceeding.

Step 2: Calculate \( \frac{A + B}{2} \) and \( \frac{A - B}{2} \)

First, find \( \frac{A + B}{2} \): \[ \frac{9x + 2x}{2} = \frac{11x}{2}. \] Next, calculate \( \frac{A - B}{2} \): \[ \frac{9x - 2x}{2} = \frac{7x}{2}. \] These values are essential in applying the identity.

Step 3: Apply the sum-to-product identity

Substitute the values from Step 2 into the sum-to-product identity: \[ \cos 9x + \cos 2x = 2 \cos \left( \frac{11x}{2} \right) \cos \left( \frac{7x}{2} \right). \] This shows the sum of cosines has been successfully expressed as a product.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are special trigonometric identities that transform the sum or difference of trigonometric functions into products. This transformation can simplify the work significantly when solving complex trigonometric equations or integrals.
In the example of \( \cos 9x + \cos 2x \), the sum of two cosine functions can be expressed as a product using the identity:

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

Identifying the angles \( A \) and \( B \) is the first crucial step. Here, \( A = 9x \) and \( B = 2x \). This identity is handy in simplifying trigonometric expressions and equations.

Cosine Addition

The cosine addition identity is an essential tool in trigonometry. It is employed in different operations, including breaking down or building up expressions.
The specific form of cosine addition used in sum-to-product identities enables us to split or transform sums into products, optimizing and simplifying calculations. By understanding this concept, you can manipulate expressions like the one given, \( \cos 9x + \cos 2x \), by employing substitution methods.

• REDUCE: Convert complicated expressions into simpler multiplication forms.
• SIMPLIFY: Make integration or solving easier in calculus applications.

Therefore, mastering these identities helps streamline solving algebraic and calculus problems.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are the fundamental building blocks of trigonometry. They are periodic functions that describe the relationships between the angles and sides of triangles in a unit circle.
The cosine function, denoted as \( \cos \theta \), is pivotal when dealing with angles and computations involving right-angled triangles. Its properties like periodicity and evenness make it ideal for various applications, not only in geometry but also in oscillatory behaviors in physics and engineering.
In our exercise, the goal was to write \( \cos 9x + \cos 2x \) as a product using known identities to simplify the expression significantly. Understanding these core trigonometric functions lays the foundation for working with more advanced mathematical concepts.

Algebraic Manipulation

Algebraic manipulation involves rearranging formulas and expressions to make calculations easier. In trigonometry, manipulation is often needed to apply identities correctly.
To solve the given problem, \( \cos 9x + \cos 2x \), knowing how to manipulate the expressions algebraically is necessary to utilize the sum-to-product identities effectively.

• EXPRESSION SUBSTITUTION: Replace components of the equation accordingly.
• SOLVE SIMPLER FORMS: Once converted, these equations are easier to handle analytically.

Thus, algebraic manipulation provides the means to transform complex trigonometric expressions into manageable forms, essential for advanced math and physics endeavors.