Problem 11 - Algebra and Trigonometry

Question

express each sum or difference as a product. If possible, find this product’s exact value. $$ \sin 7 x-\sin 3 x $$

Step-by-Step Solution

Answer

The expression \( \sin 7x - \sin 3x \) can be expressed as \( 2 \cos(5x) \sin(2x) \).

Step 1: Recognize the formula and plan the problem

Recognizing the expression form \( \sin a - \sin b \) allows us to use the trigonometric identity \( \sin a - \sin b = 2 \cos \left( \frac{a+b}{2} \right) \sin \left( \frac{a-b}{2} \right) \). Here, we can think of \( 7x \) as \( a \) and \( 3x \) as \( b \).

Step 2: Apply Trigonometric Identity

We substitute \( a \) and \( b \) into the identity, giving us: \( \sin 7x - \sin 3x = 2 \cos \left( \frac{7x+3x}{2} \right) \sin \left( \frac{7x-3x}{2} \right) \).

Step 3: Simplify the Expression

Calculating the expressions inside the brackets simplifies the expression to \( 2 \cos(5x) \sin(2x) \).

Key Concepts

Product-to-Sum Formula

Understanding the product-to-sum formula is a significant step in simplifying expressions involving trigonometric functions. This formula converts the sum or difference of sines or cosines into a product of these functions. In the expression \( \sin a - \sin b \), the product-to-sum identity used is:

• \( \sin a - \sin b = 2 \cos \left( \frac{a+b}{2} \right) \sin \left( \frac{a-b}{2} \right) \)

By recognizing our original expression \( \sin 7x - \sin 3x \) as fitting this form, we can effectively apply this identity to transition from a difference format to a product format. This not only simplifies complex trigonometric equations but also aids in solving them by potentially grounding them into more manageable forms.
This method can further simplify understanding and solving problems that involve finding the product's exact values, which might not be easily apparent in their initial formats.

Sine Function

The sine function is one of the fundamental building blocks in trigonometry, crucial for understanding oscillatory movements and waves. It is defined for any angle and typically represented as \( \sin(\theta) \), where \( \theta \) is the angle. The graph of the sine function is a continuous wave that oscillates between -1 and 1, with a period of \( 2\pi \).

• It is an odd function, meaning that \( \sin(-x) = -\sin(x) \).
• The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.

Using the product-to-sum identity transforms complex trigonometric expressions by shifting the sine terms into a more workable form for certain calculations, as seen when turning \( \sin 7x - \sin 3x \) into a product of sine and cosine. This highlights the flexibility of the sine function in mathematical transformations.

Cosine Function

The cosine function is another pivotal concept in trigonometry and forms a complementary part to the sine function. Often denoted as \( \cos(\theta) \), it is central to calculations involving angles and periodic phenomena. Just like the sine, the cosine function is defined for every angle, and its graph is a wave oscillating between -1 and 1 with a period of \( 2\pi \).

• It is an even function, satisfying \( \cos(-x) = \cos(x) \).
• In a right triangle, cosine is the ratio of the length of the adjacent side to the hypotenuse.

In the context of transforming the expression \( \sin 7x - \sin 3x \), the product-to-sum formula introduces \( \cos(5x) \), illustrating how cosine is intertwined with sine in various trigonometric identities. Recognizing the roles of both functions enhances the ease of solving and simplifying trigonometric expressions effectively.