Problem 95 - Precalculus with Limits

Question

In Exercises 91-98, use the sum-to-product formulas to write the sum or difference as a product. \( \sin (\alpha + \beta) - \sin (\alpha - \beta) \)

Step-by-Step Solution

Answer

\(\sin (\alpha + \beta) - \sin (\alpha - \beta) = 2 \cos(\beta) \sin(\alpha)\)

Step 1: Identify the pattern

Recognize that \( \sin (\alpha + \beta) - \sin (\alpha - \beta) \) follows the pattern of the sum and difference sin formula, where \( \alpha \) and \( \beta \) are the two angles.

Step 2: Apply the sum-to-product formula for sine

Apply the sine sum-to-product formula to the given expression \( \sin (\alpha + \beta) - \sin (\alpha - \beta) \). The formula is: \( \sin (\alpha + \beta) - \sin (\alpha - \beta) = 2 \cos(\beta) \sin(\alpha) \).

Step 3: Simplify the equation

Replace \( \alpha + \beta \) and \( \alpha - \beta \) in the original equation using the sine sum-to-product formula. The resulting equation will be \( 2 \cos(\beta) \sin(\alpha) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics, especially when dealing with angle transformations and simplifications. These identities are equations that involve trigonometric functions and are true for any value of the involved variables. They help us relate different trigonometric functions, enabling us to simplify complex expressions or solve trigonometric equations.

One important set of trigonometric identities is the **sum-to-product formulas**. These formulas transform the sum or difference of trigonometric functions into a product, making them easier to handle. For example, for sine functions, the formula is:

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

By using these identities, we can refactor trigonometric expressions into more manageable forms, facilitating problem-solving in trigonometry.

Sine Function

The sine function is one of the fundamental trigonometric functions, often abbreviated as \( \sin \). It is a periodic function that describes the y-coordinate of a point on the unit circle as it moves around the circle. The sine of an angle \( \theta \) can be understood as the vertical component of the radius intersecting the unit circle.

Sine functions have properties:

• They have a range between -1 and 1.
• They are periodic with a period of \( 2\pi \).
• Their graph is a continuous and smooth wave that oscillates above and below the horizontal axis.

In trigonometry, the sine function is often used to model wave-like phenomena, such as sound waves or tidal patterns. Understanding the sine function and its properties allows students to effectively apply trigonometric concepts to analyze and interpret various real-world situations.

Trigonometry

Trigonometry is a branch of mathematics that focuses on the studies of relationships between side lengths and angles in triangles, specifically right-angled triangles. It extends to concepts in circles and periodic functions.

Key areas of trigonometry include:

• Defining and using trigonometric functions such as sine, cosine, and tangent.
• Applying trigonometric identities to simplify expressions and solve equations.
• Analyzing and interpreting periodic phenomena using trigonometric models.

Trigonometry is not only important in pure mathematics but also finds applications in science, engineering, and technology. It helps in designing structures, understanding waves, and even in finding distances in astronomy. By grasping trigonometric concepts, students gain valuable tools for solving both theoretical and practical problems encountered in various fields of science and mathematics.