Problem 12 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \sin 11 x-\sin 5 x $$

Step-by-Step Solution

Answer

The expression \(\sin 11 x-\sin 5 x\) can be expressed as the product \(2\cos 8x \sin 3x\).

Step 1: Replace the given difference using Sum-to-Product Identity

Using the identity \( \sin A - \sin B = 2\cos \frac{A + B}{2}\sin \frac{A - B}{2} \), the difference can be rewritten as: \( \sin 11x -\sin 5x = 2\cos \frac{11x + 5x}{2}\sin\frac{11x - 5x}{2} \).

Step 2: Simplify the New Expression

The trigonometric expression can be simplified further to: \( 2\cos 8x\sin 3x \).

Key Concepts

Sum-to-Product Identity

Sum-to-Product identities are key tools in trigonometry that help convert sums or differences of trigonometric functions like sines and cosines into products. This is particularly useful when simplifying or solving trigonometric expressions and equations. In this exercise, we focus on the identity for the difference of sines:

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

This identity is helpful because it transforms the subtraction of two sine functions into a multiplication of cosine and sine, which can often be more manageable. For example, when given the problem \( \sin 11x - \sin 5x \), we apply the identity by considering:

• \( A = 11x \) and \( B = 5x \)

Plug these values into the identity:

• \( \sin 11x - \sin 5x = 2\cos \frac{11x + 5x}{2}\sin \frac{11x - 5x}{2} \)
• This simplifies to \( 2\cos 8x\sin 3x \)

The result is that we have successfully transformed the expression using the Sum-to-Product identity, making it easier to work with.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing the complexity of an equation or expression, often making it easier to understand or solve. In our case, we start with a complex sum or difference of functions and transform it into a simpler product format.
In step 1, we use the Sum-to-Product identity to aid in this simplification. Initially, we had \( \sin 11x - \sin 5x \). By applying the identity, we simplified it to \( 2\cos 8x\sin 3x \).

• Step-by-step simplification often involves using identities to reformat expressions, reducing terms, or even factoring where possible.

Once in product form, \( 2\cos 8x\sin 3x \) is less complex and may provide further insight for solving equations or performing calculations involving these trigonometric functions. Simplification enhances comprehension and utility in various applications.

Difference of Sines

The difference of sines formula, \( \sin A - \sin B = 2\cos \frac{A + B}{2}\sin \frac{A - B}{2} \), is particularly useful when needing to switch from a subtraction of sine functions to a product of trigonometric functions.
Understanding this transformation is crucial for solving trigonometry problems that require expressions to be multiplied instead of subtracted, especially when evaluating integrals or solving equations.

• This transformation helps in finding exact values of expressions or aids in further mathematical operations.

For instance, if you want to simplify \( \sin 11x - \sin 5x \), using this formula directly helps in reducing the problem to a more straightforward product of terms, \( 2\cos 8x\sin 3x \).
Being proficient with this identity and similar ones expands your toolkit for approaching a wide array of trigonometric problems, allowing for efficient simplification and problem-solving.