Problem 56 - Algebra and Trigonometry

Question

\(55-60\) Write the product as a sum. $$ \sin x \sin 5 x $$

Step-by-Step Solution

Answer

\( \sin x \sin 5x = \frac{1}{2} \cos(4x) - \frac{1}{2} \cos(6x) \)

Step 1: Identify the Trigonometric Identity Needed

The task is to express the product \( \sin x \sin 5x \) as a sum. We need a trigonometric identity that relates the product of two sine functions to a sum. The appropriate identity is:\[ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \] where \( A = x \) and \( B = 5x \).

Step 2: Substitute Values into the Identity

We substitute \( A = x \) and \( B = 5x \) into the identity:\[ \sin x \sin 5x = \frac{1}{2} [\cos(x-5x) - \cos(x+5x)] \] This simplifies to:\[ \sin x \sin 5x = \frac{1}{2} [\cos(-4x) - \cos(6x)] \]

Step 3: Simplify Using Cosine Identity

Recall that \( \cos(-\theta) = \cos(\theta) \), which implies:\[ \cos(-4x) = \cos(4x) \] Therefore, the expression simplifies further to:\[ \sin x \sin 5x = \frac{1}{2} [\cos(4x) - \cos(6x)] \]

Step 4: Write Final Expression

With no further simplification needed, rewrite the expression as:\[ \sin x \sin 5x = \frac{1}{2} \cos(4x) - \frac{1}{2} \cos(6x) \] This is the product written as a sum.

Key Concepts

Product-to-Sum Formulas

The product-to-sum formulas are a set of trigonometric identities that allow us to convert the product of trigonometric functions into a sum or difference. These identities are especially useful in simplifying expressions and solving equations. Let's focus on the one relevant to our exercise: the product of two sine functions.In our problem, we started with the expression \( \sin x \sin 5x \). To transform this product into a sum, we used a specific product-to-sum formula:

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

Here, \( A \) and \( B \) are the angles of the sine functions. By substituting \( A = x \) and \( B = 5x \), we can express the product \( \sin x \sin 5x \) as a sum of cosines. This technique is not only neat but it also simplifies calculations by reducing the complexity of trigonometric expressions.Understanding these transformations is crucial in higher-level mathematics, especially in calculus and engineering.

Sine Function

The sine function is one of the primary functions in trigonometry, abbreviated as \( \sin \). It is defined in the context of a right triangle as the ratio of the length of the side opposite the angle to the hypotenuse. For angles measured in radians, the sine function is periodic with a period of \( 2\pi \), which means it regularly repeats its values every \( 2\pi \) radians.

The sine function is continuous and smooth, showing up often in wave patterns, such as sound waves or light waves. In our exercise, we are specifically dealing with the products of sine functions: \( \sin x \sin 5x \). The beauty of trigonometric identities, such as the product-to-sum formulas, lies in their ability to convert complex sine products into simpler, more manageable terms like sums of cosine.

By getting comfortable with the behavior and properties of sine, including its symmetry and periodic nature, we can solve trigonometric equations more effectively.

Cosine Function

The cosine function, (\( \cos \)), is another fundamental trigonometric function. It represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle. Like the sine function, cosine also has a period of (\( 2\pi \)), meaning it repeats every (\( 2\pi \)) radians.

In the context of our problem, the product of sines is converted into a sum of cosines using the identity (\[ \cos(A-B) \text{ and } \cos(A+B) \]). The properties of cosine, especially the even nature where (\( \cos(-\theta) = \cos(\theta)\)), play a key role in simplifying expressions. For instance, knowing that (\( \cos(-4x) = \cos(4x)\)) allows us to simplify the expression without changing its value.

With a solid understanding of cosine's properties, one can not only simplify trigonometric expressions but also gain insights into real-world phenomena modeled by waves, rotations, and oscillations.