Problem 9 - Precalculus with Calculus Previews

Question

Use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ 2 \cos 3 \beta \sin \beta $$

Step-by-Step Solution

Answer

The expression is \( 2 \cos 2\beta - 2 \cos 4\beta \).

Step 1: Recall the Product-to-Sum Formulas

The Product-to-Sum identities are useful for converting the product of trigonometric functions into a sum or difference. Specifically, the formula for \( a \cos A \sin B \) is \( a \left( \cos(A-B) - \cos(A+B) \right) \). In this case, \( a = 1 \), \( A = 3\beta \), and \( B = \beta \).

Step 2: Identify Components of the Formula

Assign \( A = 3\beta \) and \( B = \beta \) to the formula \( a \cos A \sin B = a \left( \cos(A-B) - \cos(A+B) \right) \). We substitute these values into the formula.

Step 3: Apply the Product-to-Sum Formula

Using \( A = 3\beta \) and \( B = \beta \), plug in these values into the formula: \( 2 \cos 3\beta \sin \beta = 2 \left( \cos(3\beta - \beta) - \cos(3\beta + \beta) \right) \).

Step 4: Simplify the Expression

Simplify the expression from Step 3: \( 2 \cos(3\beta - \beta) - 2 \cos(3\beta + \beta) = 2 \cos 2\beta - 2 \cos 4\beta \).

Step 5: Write the Final Answer

The expression \( 2 \cos 3\beta \sin \beta \) as a sum of cosines is \( 2 \cos 2\beta - 2 \cos 4\beta \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all angles and are used extensively in solving trigonometry problems. These identities simplify complex trigonometric expressions and enable us to relate different trigonometric functions to one another.
One crucial set of identities involves relating products to sums, which are handy when working with angles in trigonometric functions. Among these are the Product-to-Sum formulas, which transform products of sine and cosine functions into sum or difference expressions.
Understanding these identities is essential as they can simplify the computation and manipulation of trigonometric equations. When applying trigonometric identities, it's important to remember:

• They apply to all angles, making them universally useful.
• Memorization can be helpful, but understanding the derivation and application is key to using them effectively.

Sum and Difference of Angles

The sum and difference of angles formulas are fundamental in trigonometry because they allow us to express trigonometric functions of sums or differences of angles in terms of the functions of individual angles. These include:

• \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
• \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)

These formulas are incredibly useful in reducing complex trigonometric expressions into simpler forms or in cases where you need to solve for a specific angle. In the context of the product-to-sum formulas, identifying and isolating these components allows the transformation from a product of functions into a sum or difference, as demonstrated by the step-by-step solution of the given problem.
The application of these formulas requires recognizing patterns in trigonometric expressions that fit these identities. Mastering sum and difference identities is crucial, especially when solving equations that initially appear daunting.

Trigonometry Problem Solving

Solving trigonometry problems often involves a combination of understanding identities, applying formulas, and manipulating expressions skillfully. When faced with a trigonometric product, like \( 2 \cos 3\beta \sin \beta \), utilizing the product-to-sum formula streamlines the process significantly.
Here's a summary of the problem-solving steps employed in the exercise solution:

• Identify the appropriate identity or formula that applies to the situation. In this case, it's the product-to-sum formula for \( a \cos A \sin B \).
• Substitute the specific values for \( A \) and \( B \) in the formula to transform the product into a sum or difference expression.
• Simplify the resulting expression step-by-step, ensuring each transformation is consistent with trigonometric rules.
• By simplifying \( 2 \left( \cos(3\beta - \beta) - \cos(3\beta + \beta) \right) \), it becomes \( 2 \cos 2\beta - 2 \cos 4\beta \), a direct solution in terms of sum expressions.

When tackling any trigonometry problem, break down the issue into smaller parts, apply the right formulas, and verify each step. Patience and practice will lead to confident and accurate trigonometry problem solving.