Problem 57 - Precalculus Essentials

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: Trigonometric Identities

Starting point is to recall the identities for \( \sin (\alpha+\beta) \) and \( \sin (\alpha-\beta) \). According to the sum and difference identities we have, \( \sin (\alpha+\beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \) and \( \sin (\alpha-\beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

Step 2: Subtracting the Identities

Next, subtract the identity for \( \sin (\alpha-\beta) \) from the identity for \( \sin (\alpha+\beta) \). We have, \( \sin (\alpha+\beta) - \sin (\alpha-\beta)= [\sin \alpha \cos \beta + \cos \alpha \sin \beta ] - [\sin \alpha \cos \beta - \cos \alpha \sin \beta ] \) or, \( 2 \cos \alpha \sin \beta = \sin (\alpha+\beta) - \sin (\alpha-\beta) \)

Step 3: Rearrange for the Final Formula

Now we will rearrange above resulting formula to make \( \cos \alpha \sin \beta \) subject of the formula. This will result in the product-to-sum formula for \( \cos \alpha \sin \beta \). So, \( \cos \alpha \sin \beta = \frac{1}{2} [\sin (\alpha+\beta) - \sin (\alpha-\beta) ] \)

Key Concepts

Sum and Difference Formulas

Sum and difference formulas are crucial tools in trigonometry. They allow us to find the sine, cosine, and tangent of the sum or difference of two angles. For sine, the formulas are:

• For the sum of two angles: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• For the difference of two angles: \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

These formulas can be incredibly helpful for solving trigonometric equations and simplifying expressions. By understanding them, you can break down complex trigonometric problems into simpler, more manageable pieces.
Once mastered, these identities also assist in deriving other important trigonometric formulas, such as product-to-sum identities.
It's essential to memorize these formulas as they lay the groundwork for many advanced trigonometric concepts.

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences. This conversion can simplify the integration and calculation of expressions. One such formula, derived in the step-by-step solution is:

• \( \cos \alpha \sin \beta = \frac{1}{2} [\sin(\alpha + \beta) - \sin(\alpha - \beta)] \)

This formula allows us to relate the product \( \cos \alpha \sin \beta \) to a difference of sines, simplifying numerous trigonometric computations.
Using product-to-sum formulas can make complex calculus problems more approachable, offering a method to convert difficult multiplication into easier addition or subtraction problems.
Understanding these connections can help you manipulate trigonometric expressions more flexibly, broadening your problem-solving toolbox.

Sine and Cosine Identities

Sine and cosine identities are foundational in trigonometry. They describe relationships between angles and their sine or cosine values, allowing for profound insights into geometric and trigonometric problems. Some identities include:

• The Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Complementary angle identities: \( \sin(90^\circ - \theta) = \cos \theta \) and \( \cos(90^\circ - \theta) = \sin \theta \)

These identities help verify solutions and transform expressions, offering many ways to approach sinusoidal problems.
They also provide critical insights for calculus, specifically in terms of differentiating and integrating trigonometric functions.
By mastering sine and cosine identities, you lay a solid foundation for further exploration in mathematics, abstract algebra, and even physics.