Problem 10 - Precalculus Essentials

Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\sin 8 x+\sin 2 x$$

Step-by-Step Solution

Answer

The expression \(\sin 8x + \sin 2x\) can then be rewritten as \(2 \sin 5x \cos 3x\)

Step 1: Identifying the Terms

The given expression is \(\sin 8x + \sin 2x\). Here, A and B would be considered as 8x and 2x respectively.

Step 2: Applying the Sum-to-product Trigonometric Identity

The sum-to-product identity \(\sin A + \sin B = 2 \sin \frac{1}{2}(A+B) \cos \frac{1}{2}(A-B)\) can be applied here. Replacing A with 8x and B with 2x, the expression becomes: \n2 \sin \frac{1}{2}((8x+2x)) \cos \frac{1}{2}((8x-2x))

Step 3: Simplify the Expression

Simplify the expression inside the sin and cos functions.\nThis becomes: 2 \sin \frac{1}{2}(10x) \cos \frac{1}{2}(6x), which simplifies to: \n2 \sin 5x \cos 3x.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are a handy part of trigonometry. They allow you to turn sums into products, which can often simplify complex expressions or make them easier to solve. A sum-to-product identity gives a way to express trigonometric expressions in terms of products instead of sums or differences.
In the original exercise, we are using one of these identities:

• \( \sin A + \sin B = 2 \sin \frac{1}{2}(A+B) \cos \frac{1}{2}(A-B) \)

When faced with an expression like \( \sin 8x + \sin 2x \), identifying it as a candidate for a sum-to-product identity is the first step. We then apply the identity to break the sum down into a simpler product. This approach is especially useful because products can be more straightforward to compute or simplify further. Understanding these identities can make navigating through trigonometric problems more manageable.

Trigonometric Functions

Trigonometric functions such as sine and cosine are central to trigonometry. In the exercise, both functions are utilized in the identity for transforming a sum into a product.
Sine and cosine functions help describe the relation between side lengths and angles in right triangles. They function within a certain range: sine ranging from -1 to 1 and cosine also from -1 to 1. They are periodic, repeating their values in regular intervals, which makes them useful for modeling wave-like phenomena.
In the expression \( \sin 8x + \sin 2x \), we see two angles, 8x and 2x. When using the sum-to-product identity, these angles are manipulated into new arguments for the sine and cosine functions. Notably,

• The angle \( A+B = 10x \) sets the argument for the sine function.
• The angle \( A-B = 6x \) defines the argument for the cosine function.

Aligned with the identity, these equations help create a new expression, reshaping the original challenge into something simplified or more directly solvable.

Problem Solving

Problem-solving in trigonometry often involves transforming expressions using a variety of mathematical identities. Knowing which identity to apply and how to manipulate angles is crucial.
In the example of \( \sin 8x + \sin 2x \), the problem-solving process involves several key steps:

• Identifying that the given expression can be simplified using a sum-to-product identity.
• Applying the appropriate identity, transforming the sum into a product expression.
• Simplifying the resulting expression into a more manageable form, such as \( 2 \sin 5x \cos 3x \).

Each step builds upon the previous, requiring understanding of both the identity used and how the trigonometric functions interact. This clear pathway to a solution helps ensure that complex expressions can be handled with confidence and accuracy. Practicing such problem-solving techniques is essential for mastering trigonometry.