Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the product as a sum or difference. \(6 \sin 8 \theta \cos 3 \theta\)

Step-by-Step Solution

Answer

The product \(6 \sin 8 \theta \cos 3 \theta\) as a sum is \(3 \sin 11\theta + 3 \sin 5\theta\).

Step 1: Identify the trigonometric identity

The trigonometric identity to use here is the product-to-sum identity: \(2 \sin A \cos B = \sin(A+B) + \sin(A-B)\)

Step 2: Apply the product-to-sum identity

Rewrite the expression using the product-to-sum identity. Notice that there is a coefficient of 6 in the original expression, which is divided by 2 as a result of the identity application: \(6 \sin 8 \theta \cos 3 \theta = 3[\sin(8\theta+3\theta)+\sin(8\theta-3\theta)]\)

Step 3: Simplify the expression

Simplify the expression to achieve the sum form: \(3 \sin 11\theta + 3 \sin 5\theta\)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. Among these useful identities are the product-to-sum identities. These identities help us transform a product of trigonometric functions, like sine and cosine, into a sum or difference. This transformation makes it easier to handle and integrate these expressions in various mathematical problems.

For example, when you have an expression like \(6 \sin 8\theta \cos 3\theta\), the product-to-sum identity can be applied to simplify it. The identity \(2 \sin A \cos B = \sin(A+B) + \sin(A-B)\) is particularly helpful here. Applying this, we break down the multiplication into addition which can simplify the process of solving the problem or integrating the equation later on. Recognizing and applying the right identity is crucial to making complex trigonometric expressions manageable.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry. They describe the relationships between the angles and sides of a right-angled triangle, but they also extend into functions that model periodic phenomena, like sound and light waves.

Understanding how sine and cosine interact is essential, especially when dealing with combined angles. In the expression \(6 \sin 8\theta \cos 3\theta\), the sine and cosine functions are being multiplied together, indicating a need for transformation using identities. When we break these interactions into sums or differences, it visually and practically simplifies the functions' complexity. For instance, by using the product-to-sum identities, these two functions help in simplifying the expression to \(3 \sin 11\theta + 3 \sin 5\theta\), making it much more straightforward to analyze.

Simplifying Expressions

Simplifying expressions in mathematics means to reduce them to their most concise and understandable form. This often involves using identities, algebraic manipulations, and strategic transformations of mathematical expressions.

In our specific exercise, simplifying \(6 \sin 8\theta \cos 3\theta\) involves recognizing that it's a product that can be simplified using a product-to-sum identity. We multiply the identity \(2 \sin A \cos B = \sin(A+B) + \sin(A-B)\) by \(3\) to directly simplify the given product into a sum: \(3[\sin(11\theta) + \sin(5\theta)]\). This simplification process is invaluable because it presents a more compact form of the original expression. It aids in further mathematical operations like integration, differentiation, or evaluating expressions at particular angles.