Problem 57 - Algebra and Trigonometry

Question

Use the identities for \(\sin (\alpha+\beta)\) and \(\sin (\alpha-\beta)\) to solve Subtract the left and right sides of the identities and derive the product-to- sum formula for \(\cos \alpha \sin \beta\)

Step-by-Step Solution

Answer

The product-to-sum formula for \(\cos \alpha \sin \beta\) is \(\cos{\alpha}\sin{\beta} = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)]\).

Step 1: Write Down the Sine Sum and Difference Identities

The identities for \(\sin (\alpha+\beta)\) and \(\sin (\alpha-\beta)\) should be written down first. These identities are typically given as: \(\sin (\alpha+\beta) = \sin{\alpha}\cos{\beta} + \cos{\alpha}\sin{\beta}\) and \(\sin (\alpha-\beta) = \sin{\alpha}\cos{\beta} - \cos{\alpha}\sin{\beta}\)

Step 2: Subtract the Identities

Now subtract the equation for \(\sin (\alpha-\beta)\) from the equation for \(\sin (\alpha+\beta)\). As a result, we get: \(\sin (\alpha+\beta) - \sin (\alpha-\beta) = [\sin{\alpha}\cos{\beta} + \cos{\alpha}\sin{\beta}] - [\sin{\alpha}\cos{\beta} - \cos{\alpha}\sin{\beta}]\)

Step 3: Simplify the Expression

After subtracting the right sides of the identities, we get: \(\sin (\alpha+\beta) - \sin (\alpha-\beta) = 2\cos{\alpha}\sin{\beta}\)

Step 4: Derive the Product-to-Sum Formula for \(\cos \alpha \sin \beta\)

To isolate \(\cos{\alpha}\sin{\beta}\) on one side and get the formula we need, divide the whole equation by 2: \(\cos{\alpha}\sin{\beta} = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)]\). This is the product-to-sum formula for \(\cos \alpha \sin \beta\)

Key Concepts

Sine Sum and Difference Identities

Understanding sine sum and difference identities is crucial for delving into more complex trigonometric equations. These identities tie together the sine of a sum or difference of angles to the product of sines and cosines of individual angles. Specifically, the sum identity is given by
\[\sin (\alpha+\beta) = \sin{\alpha}\cos{\beta} + \cos{\alpha}\sin{\beta}\]
Similarly, the difference identity is given by
\[\sin (\alpha-\beta) = \sin{\alpha}\cos{\beta} - \cos{\alpha}\sin{\beta}\]
These identities are pivotal because they enable us to convert from sums or differences to product forms and vice versa, which often simplifies solving equations. For instance, by subtracting the difference identity from the sum identity, as shown in the exercise, we can derive product-to-sum formulas that are essential in various trigonometry applications, including integration and series expansions.

Trigonometric Identity

A trigonometric identity is an equation that holds true for all values within its domain. They are the backbone of trigonometry, serving as tools to simplify expressions, prove other identities, and solve trigonometric equations. One common example of a trigonometric identity that we used in deriving the product-to-sum formula is
\[\sin^2(\alpha) + \cos^2(\alpha) = 1\]
The product-to-sum formula itself becomes a helpful identity:
\[\cos{\alpha}\sin{\beta} = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)]\]
It is important to recognize identities such as this during problem-solving, as they can transform seemingly complex problems into more manageable ones. Committing the most frequently used identities to memory pays dividends when quickly navigating through trigonometry problems.

Solving Trigonometric Equations

The domain of trigonometry includes various types of equations that involve trigonometric functions. Solving these equations typically requires one to manipulate the functions using identities to isolate the variable of interest. The process often involves algebraic manipulation and can include the use of sum-to-product and product-to-sum formulas. For example, in our exercise, we isolated \(\cos{\alpha}\sin{\beta}\) by employing the derived product-to-sum identity. In more complex cases, you might need to employ additional strategies such as squaring both sides, factoring, or utilizing co-function identities. Being adept at using these tools is essential for solving trigonometric equations effectively.

Algebraic Manipulation in Trigonometry

Algebraic manipulation in trigonometry is much like it is in algebra: it involves rearranging terms, factoring, and simplifying expressions to get a desired outcome. These fundamental skills aid in the transformation of trigonometric expressions and the solving of trigonometric equations. For example, in deriving the product-to-sum formula, algebraic manipulation was key. We subtracted one identity from the other and then divided by 2 to isolate the term \(\cos{\alpha}\sin{\beta}\).

Throughout your study of trigonometry, you'll find that algebraic manipulation can make complex problems simpler to understand and solve. It's an overlap of disciplines that showcases how interconnected different areas of mathematics truly are.