Problem 45 - Algebra and Trigonometry

Question

41–46 Write the product as a sum. $$3 \cos 4 x \cos 7 x$$

Step-by-Step Solution

Answer

\(3 \cos 4x \cos 7x = \frac{3}{2} \cos(11x) + \frac{3}{2} \cos(3x)\) as a sum.

Step 1: Identify the Trigonometric Product Formula

To express the product of two cosine functions as a sum, we use the product-to-sum formula for cosine:\[ \cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)] \]In this exercise, identify \(A = 4x\) and \(B = 7x\).

Step 2: Apply the Product-to-Sum Formula

Substitute \(A = 4x\) and \(B = 7x\) into the formula:\[ \cos 4x \cos 7x = \frac{1}{2} [\cos(4x + 7x) + \cos(4x - 7x)] \]This simplifies to:\[ \cos 4x \cos 7x = \frac{1}{2} [\cos(11x) + \cos(-3x)] \].

Step 3: Simplify the Expression

Remember that \(\cos(-\theta) = \cos(\theta)\). Therefore, \(\cos(-3x) = \cos(3x)\). Substituting this back into the equation gives:\[ \cos 4x \cos 7x = \frac{1}{2} [\cos(11x) + \cos(3x)] \].

Step 4: Multiply by the Coefficient

The initial problem multiplies the cosine product by 3, so multiply the entire expression by 3:\[ 3 \cos 4x \cos 7x = 3 \times \frac{1}{2} [\cos(11x) + \cos(3x)] \]This results in:\[ 3 \cos 4x \cos 7x = \frac{3}{2} \cos(11x) + \frac{3}{2} \cos(3x) \].

Key Concepts

Product-to-Sum Formulas

The product-to-sum formulas in trigonometry are an essential tool for simplifying the product of trigonometric functions into a sum or difference of simpler trigonometric terms. These formulas are especially useful when dealing with problems involving products of sine and cosine functions, as they allow for easier manipulation and simplification.

The specific product-to-sum formula for cosine is given by: \[ \cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)] \].

This formula transforms the product of two cosine functions into a sum of two cosine functions. Here are a few tips for using product-to-sum formulas effectively:

• Ensure you correctly identify the angles (or variables) for A and B.
• Substitute your values accurately into the formula.
• Apply algebraic simplifications as needed, such as handling negative angles.

Practicing these formulas can greatly aid in handling complex trigonometric expressions that initially seem daunting.

Cosine Functions

Cosine functions are one of the fundamental trigonometric functions often encountered in trigonometry exercises and practical applications. The cosine function, usually denoted as \( \cos(\theta) \), gives the x-coordinate of a point on the unit circle corresponding to a given angle \( \theta \). Some key points about cosine functions include:

• Cosine functions are periodic with a period of \( 2\pi \), meaning they repeat their values in every interval of \( 2\pi \).
• The cosine function is even, which means \( \cos(-\theta) = \cos(\theta) \).
• It ranges from -1 to 1, with maximum value 1 when \( \theta = 0, 2\pi, 4\pi, \ldots \) and minimum value -1 when \( \theta = \pi, 3\pi, 5\pi, \ldots \).

When using cosine in trigonometric identities, be aware of these properties to simplify and solve expressions easily.

Trigonometric Simplification

Trigonometric simplification plays a crucial role in solving many mathematical problems efficiently. By simplifying trigonometric expressions, you can make computations easier and highlight relationships between various mathematical elements.

Here are some approaches and tips to help simplify trigonometric expressions:

• Use known identities such as Pythagorean, reciprocal, and quotient identities to transform expressions into simpler forms.
• Apply algebraic techniques such as factoring and distributing to make expressions more manageable.
• Combine and rearrange terms strategically to reveal simplification opportunities, such as using symmetry or periodic properties of trigonometric functions.

Approaching trigonometric problems with a focus on simplification can reduce complexity and help uncover the core insights needed to reach a solution.