Problem 41 - Algebra and Trigonometry

Question

41–46 Write the product as a sum. $$\sin 2 x \cos 3 x$$

Step-by-Step Solution

Answer

\(\sin 2x \cos 3x = \frac{1}{2} (\sin 5x - \sin x)\).

Step 1: Identify the Product-to-Sum Formula

We need to apply a trigonometric identity to rewrite the product of sine and cosine as a sum. The relevant identity is: \( \sin A \cos B = \frac{1}{2}(\sin(A+B) + \sin(A-B)) \). Here, \( A = 2x \) and \( B = 3x \).

Step 2: Apply the Formula

Substitute \( A = 2x \) and \( B = 3x \) into the formula: \\[ \sin 2x \cos 3x = \frac{1}{2}(\sin((2x + 3x)) + \sin((2x - 3x))). \]

Step 3: Simplify the Expression

Calculate \( 2x + 3x = 5x \) and \( 2x - 3x = -x \). Therefore, the expression becomes: \\[ \sin 2x \cos 3x = \frac{1}{2}(\sin 5x + \sin(-x)). \]

Step 4: Use the Odd Property of Sine Function

Since \( \sin(-x) = -\sin x \), we can further simplify: \\[ \sin 2x \cos 3x = \frac{1}{2}(\sin 5x - \sin x). \]

Key Concepts

Product-to-sum formula

The product-to-sum formula is a useful tool in trigonometry, especially when you need to convert products of trigonometric functions into summation forms. This transformation often simplifies the problem at hand. The general idea is that when you have an expression involving the product of sine and cosine, you can apply this formula to turn it into a sum, which is generally easier to manage.

For instance, the identity that we've applied in the exercise is: \( \sin A \cos B = \frac{1}{2}(\sin(A+B) + \sin(A-B)) \). By substituting particular values for \( A \) and \( B \), it becomes a straightforward substitution exercise. This formula not only simplifies expressions but also helps in integrating or differentiating trigonometric expressions by breaking them into simpler parts.

In practical applications, converting products to sums can reveal symmetries or patterns that were not initially apparent. This step is often used in signal processing or physics where wave-type functions are common, and analyzing their behaviors benefits from such conversion.

Sine and cosine functions

Sine and cosine functions are the foundational building blocks of trigonometry. They represent periodic phenomena and are crucial in studying oscillations, waves, and circular motion. Both functions have specific properties that make them unique and widely applicable:

• Periodic Nature: Sine and cosine functions both have a period of \( 2\pi \), meaning they repeat their values every \( 2\pi \) interval.
• Range and Domain: Both functions have a range of [-1, 1] and a domain of all real numbers. This means they will oscillate between -1 and 1 for all inputs.
• Odd and Even Functions: Sine is an odd function, which means \( \sin(-x) = -\sin(x) \). Cosine is an even function, meaning \( \cos(-x) = \cos(x) \).

The interplay of these functions is significant in converting product expressions into sums or differences. Understanding these properties can lead to better comprehension of how and why these conversions and simplifications work as they do. This knowledge is not only theoretical but extensively used in fields like engineering, physics, and computer graphics.

Simplifying expressions

Simplifying trigonometric expressions can make complex problems more approachable and easier to solve. This phase often involves using trigonometric identities to re-formulate expressions into simpler or more useful forms.

For example, in the given problem, using the product-to-sum formula to convert \( \sin 2x \cos 3x \) into \( \frac{1}{2}(\sin 5x - \sin x) \) reduces the complexity by turning a product into a sum. This is advantageous when solving further mathematical problems, such as integrations or differential equations..

The simplification process often involves:

• Recognizing applicable identities, such as the product-to-sum, Pythagorean, or angle sum identities.
• Applying properties of sine and cosine, like their periodicity or symmetry, to rearrange terms effectively.
• Reducing terms step-by-step by breaking them down into fundamental components.

Every time you simplify correctly, you make the expression more "calculator-friendly" and provide clearer steps to finding solutions, whether they be algebraic or numeric. Simplifying isn't just a procedure; it's a strategic method to seeing the underlying pattern or solution more clearly.