Problem 48 - Algebra and Trigonometry

Question

47–52 Write the sum as a product. $$\sin x-\sin 4 x$$

Step-by-Step Solution

Answer

The sum can be written as \(-2 \cos\left( \frac{5x}{2} \right) \sin\left( \frac{3x}{2} \right)\).

Step 1: Identify the Formula to Use

To rewrite the expression \( \sin x - \sin 4x \) as a product, we use the sum-to-product identities. The relevant identity here is: \( \sin A - \sin B = 2 \cos\left( \frac{A+B}{2} \right) \sin\left( \frac{A-B}{2} \right) \).

Step 2: Assign A and B

Identify \( A = x \) and \( B = 4x \) in the expression.

Step 3: Calculate \( \frac{A+B}{2} \) and \( \frac{A-B}{2} \)

Calculate \( \frac{A+B}{2} = \frac{x + 4x}{2} = \frac{5x}{2} \) and \( \frac{A-B}{2} = \frac{x - 4x}{2} = \frac{-3x}{2} \).

Step 4: Apply the Identity

Using the values from Step 3, apply the identity: \( \sin x - \sin 4x = 2 \cos\left( \frac{5x}{2} \right) \sin\left( \frac{-3x}{2} \right) \).

Step 5: Simplify the Expression

The sine function is odd, so \( \sin\left( \frac{-3x}{2} \right) = -\sin\left( \frac{3x}{2} \right) \). Simplifying the expression gives: \( 2 \cos\left( \frac{5x}{2} \right) \cdot (-\sin\left( \frac{3x}{2} \right)) = -2 \cos\left( \frac{5x}{2} \right) \sin\left( \frac{3x}{2} \right) \).

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are essential tools in mathematics that make it easier to manipulate and simplify expressions involving trigonometric functions. These identities are equations that hold true for all values of the variables involved. Learning them can make solving complex trigonometric problems much more manageable.

Some commonly used trigonometric identities include the Pythagorean identities, angle sum and difference identities, and double-angle identities. Another important set are the sum-to-product identities. These sense to transform the sums or differences of trigonometric functions into products, which often simplifies the solution process.

• For example, the sum-to-product identity for sine states: \( \sin A - \sin B = 2 \cos\left( \frac{A+B}{2} \right) \sin\left( \frac{A-B}{2} \right) \).
• Using such identities can be incredibly useful for simplifying oscillatory functions or solving integrals involving trigonometric terms.

Understanding and practicing these identities is crucial for anyone working with trigonometry. As with any mathematical tool, the more you use them, the better you'll understand and remember them.

Exploring Sine and Cosine Functions

Sine and cosine functions are fundamental to trigonometry. They describe the relationship between the angles and sides of right-angled triangles. When dealing with these functions, it’s key to remember some of their unique properties.

• The sine function, \( \sin x \), is an odd function, meaning that \( \sin(-x) = -\sin x \).
  
• The cosine function, \( \cos x \), is an even function, meaning that \( \cos(-x) = \cos x \).

The sine function measures the opposite side over the hypotenuse in a right-angled triangle, whereas the cosine function evaluates the adjacent side over the hypotenuse. Understanding these basic properties helps in transforming and simplifying expressions involving these functions.

Together, the sine and cosine functions are vital for converting angular expressions to a more operable form, like turning a subtraction equation into a single product function. They are also periodic functions, which means they repeat their values in regular intervals. This periodic nature is crucial in fields like physics, engineering, and computer graphics.

The Art of Expression Simplification

Expression simplification in mathematics is not just reducing terms but making an expression easier to interpret or analyze. In trigonometry, simplification often involves rearranging expressions using various identities to make them more compact or to facilitate further integration or differentiation.

In the example \( \sin x - \sin 4x \), the simplification process involves several essential steps:

• Identify the right identity: Knowing which identity can effectively transform the original expression is crucial. In this case, the sum-to-product identity for sine is applied.
• Correctly assigning components: Ensure each variable matches the identity format. Here, \( A = x \) and \( B = 4x \).
• Performing accurate calculations: Compute the necessary parts, such as \( \frac{A+B}{2} \) and \( \frac{A-B}{2} \), leading to efficient application of the identities.

Eventually, the result \( -2 \cos\left( \frac{5x}{2} \right) \sin\left( \frac{3x}{2} \right) \) is a simplified form of the original subtraction, allowing for easier further mathematical manipulation or interpretation.