Problem 65 - Algebra and Trigonometry

Question

\(61-66\) Write the sum as a product. $$ \sin 2 x-\sin 7 x $$

Step-by-Step Solution

Answer

\(-2 \cos(4.5x) \sin(2.5x)\)

Step 1: Recognize the Trigonometric Identity

To solve this problem, we will use the trigonometric identity for the difference of sines: \[\sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)\]This formula allows us to express a difference of sines as a product of a cosine and a sine function.

Step 2: Substitute Values in the Identity

Here, we have \(A = 2x\) and \(B = 7x\). Substitute these values into the identity:\[\sin 2x - \sin 7x = 2 \cos \left(\frac{2x + 7x}{2}\right) \sin \left(\frac{2x - 7x}{2}\right)\]

Step 3: Simplify the Expression

Let's perform the arithmetic inside the cosine and sine functions:- \( \frac{2x + 7x}{2} = \frac{9x}{2} = 4.5x\)- \( \frac{2x - 7x}{2} = \frac{-5x}{2} = -2.5x\)Substitute these simplified values back into the expression:\[2 \cos (4.5x) \sin (-2.5x)\]

Step 4: Simplify Further Using Symmetry

The sine function is odd, meaning \( \sin(-\theta) = -\sin(\theta) \). Thus:\[2 \cos (4.5x) \sin (-2.5x) = -2 \cos (4.5x) \sin (2.5x)\]This step simplifies the expression into a consistent form.

Key Concepts

Difference of Sines

The difference of sines is a classic concept in trigonometry, used to transform expressions involving two sine functions into a product form. This transformation can make the expression easier to handle, especially when dealing with integrations or solving equations. For the difference of two sine functions, the identity:\[ \sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \]is pivotal. This identity allows us to express the difference in terms of cosine and sine products. It's useful because it often leads to simplifications in complex problems.Here’s a quick breakdown of how it works:

• Calculate the average angle \(\frac{A+B}{2}\) for the cosine term.
• Compute half the difference \(\frac{A-B}{2}\) for the sine term.

These two components transform the difference of sines into a product of cosine and sine, simplifying the original expression.

Trigonometric Formulas

Trigonometric formulas are the backbone of solving problems in trigonometry. They provide structured methods to transform or manipulate trigonometric expressions. The formulas for the difference and sum of sines or cosines are especially useful. They break expressions down in a way that can simplify solving or further manipulation.In particular, understanding identities like:\[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \]or the one used in our exercise:\[ \sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \]These identities show how an apparently complex subtraction can transform into a simpler product. This is pivotal in reducing the complexity of calculations and solving equations involving trigonometric functions. Mastery of these formulas can significantly ease the study of trigonometry.

Trigonometric Expressions

Trigonometric expressions are combinations of the basic sine, cosine, and tangent functions. They might initially seem complex, but many techniques and identities can simplify them. Harnessing these expressions effectively requires a good understanding of various transformations and simplifications.In our exercise, transforming the expression \( \sin 2x - \sin 7x \) demonstrates how using a trigonometric identity can simplify the expression into \(-2 \cos(4.5x) \sin(2.5x)\). This reduction:

• Makes the expression more manageable.
• Highlights how trigonometric identities are essential tools.
• Reveals the underlying structure and symmetry of trigonometric functions.

By breaking down complex expressions into simpler forms, we gain both a practical solution and a deeper insight into how trigonometric functions interplay. Understanding such transformations is key in higher-level mathematics and applications in physics and engineering.