Problem 65 - Precalculus Mathematics for Calculus

Question

Write the sum as a product. $$\sin 2 x-\sin 7 x$$

Step-by-Step Solution

Answer

\( \sin 2x - \sin 7x = -2 \cos \left( \frac{9x}{2} \right) \sin \left( \frac{5x}{2} \right) \).

Step 1: Identify the Trigonometric Identity

The expression \(\sin 2x - \sin 7x \) is a difference of sines. We can use the trigonometric identity for the difference of sines: \[ \sin A - \sin B = 2 \cos \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \] where \(A = 2x\) and \(B = 7x\).

Step 2: Calculate the Values for A+B and A-B

Calculate the sum and the difference of \(A\) and \(B\):\[A + B = 2x + 7x = 9x \]\[A - B = 2x - 7x = -5x \]These will be used in the identity to find the cosine and sine arguments.

Step 3: Apply the Trigonometric Identity

Using the identity from Step 1, substitute the calculated values: \[\sin 2x - \sin 7x = 2 \cos \left( \frac{9x}{2} \right) \sin \left( \frac{-5x}{2} \right) \].

Step 4: Simplify the Expression

Since sine is an odd function, we can simplify \(\sin \left( \frac{-5x}{2} \right) \) to \[-\sin \left( \frac{5x}{2} \right) \]. Thus, the expression becomes:\[\sin 2x - \sin 7x = -2 \cos \left( \frac{9x}{2} \right) \sin \left( \frac{5x}{2} \right) \].

Key Concepts

Difference of Sines Identity

The difference of sines identity is a powerful tool in trigonometry. It allows us to transform the difference of two sine functions into a product of sine and cosine. This transformation is beneficial for simplifying complex trigonometric expressions. The identity is given by:

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

To use this identity effectively, identify the angles \( A \) and \( B \), then simply calculate their sum and difference. Substitute these values into the identity to convert the expression. In our problem, with \( A = 2x \) and \( B = 7x \), we calculated \( A + B = 9x \) and \( A - B = -5x \). This substitution helps in seeing the expression in a simpler form, which is easier to work with for further calculations or integrations. Understanding and applying this identity is crucial for manipulating trigonometric equations in algebra and calculus.

Sum to Product Formulas

The sum to product formulas are essential for converting sums or differences of trigonometric functions into a product format. This is helpful in simplifying trigonometric expressions and solving equations. The formulas allow for a straightforward conversion:

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

For the given problem, we specifically used the formula for \( \sin A - \sin B \), where it is transformed into a product involving cosine and sine. The simplification not only reduces the complexity but also provides a more usable form for further mathematical processes, such as integration or differentiation in advanced mathematics studies. Recognizing when and how to use these formulas is key to mastering trigonometric transformations.

Cosine and Sine Simplification

Simplifying trigonometric expressions often involves applying properties and identities, such as those for cosine and sine functions. In our specific problem, after applying the difference of sines identity, we further simplify the expression by leveraging the fact that sine is an odd function. This property is crucial because it tells us:

• \( \sin(-x) = -\sin(x) \)

Using this, we simplified \( \sin \left( \frac{-5x}{2} \right) \) to \(-\sin \left( \frac{5x}{2} \right) \), leading us to a more straightforward form of the expression:

• \( \sin 2x - \sin 7x = -2 \cos \left( \frac{9x}{2} \right) \sin \left( \frac{5x}{2} \right) \)

This simplification step is critical as it ensures that the final expression is both accurate and easier to handle within various mathematical applications. A deep grasp of these simplifications is vital for efficiently solving more complex trigonometric problems.