Problem 46 - Algebra and Trigonometry

Question

41–46 Write the product as a sum. $$11 \sin \frac{x}{2} \cos \frac{x}{4}$$

Step-by-Step Solution

Answer

\( \frac{11}{2} \left( \sin \frac{3x}{4} + \sin \frac{x}{4} \right) \)

Step 1: Identify the Trigonometric Identity

We notice that the expression involves the product of sine and cosine functions. We can use the trigonometric product-to-sum identities to rewrite this product as a sum: \[ \sin A \cos B = \frac{1}{2} \left( \sin (A + B) + \sin (A - B) \right) \]

Step 2: Apply the Identity

Given the expression \(11 \sin \frac{x}{2} \cos \frac{x}{4}\), we can apply the product-to-sum formula identified in Step 1 with \(A = \frac{x}{2}\) and \(B = \frac{x}{4}\). Thus, it becomes \[ \sin \frac{x}{2} \cos \frac{x}{4} = \frac{1}{2} \left( \sin \left(\frac{x}{2} + \frac{x}{4}\right) + \sin \left(\frac{x}{2} - \frac{x}{4}\right) \right) \]

Step 3: Simplify the Sum of Angles

Calculate the sum and difference inside the sine functions:- Sum: \(\frac{x}{2} + \frac{x}{4} = \frac{2x}{4} + \frac{x}{4} = \frac{3x}{4}\) - Difference: \(\frac{x}{2} - \frac{x}{4} = \frac{2x}{4} - \frac{x}{4} = \frac{x}{4}\) So, we have:\[ \frac{1}{2} \left( \sin \frac{3x}{4} + \sin \frac{x}{4} \right) \]

Step 4: Multiply the Identity by the Constant

Since we have the initial factor of \(11\), we need to multiply the simplified result by this constant. Thus, the expression becomes:\[ 11 \times \frac{1}{2} \left( \sin \frac{3x}{4} + \sin \frac{x}{4} \right) = \frac{11}{2} \left( \sin \frac{3x}{4} + \sin \frac{x}{4} \right) \]

Key Concepts

product-to-sum identities

Trigonometric identities are powerful formulas used to simplify expressions involving trigonometric functions. One such useful set of identities is the product-to-sum identities. These identities allow us to convert products of trigonometric functions, like sine and cosine, into sums or differences. This is particularly handy because sums and differences are often easier to manipulate and analyze compared to products.
For example, one of the product-to-sum identities is \( \sin A \cos B = \frac{1}{2} (\sin(A+B) + \sin(A-B)) \). This can be very useful when dealing with expressions like \( 11 \sin \frac{x}{2} \cos \frac{x}{4} \).

• This identity helps in breaking down the problem into simpler parts.
• You substitute directly into the identity to transform the product into a sum.

Understanding these identities can greatly aid in trigonometric simplification, making complex expressions much more manageable.

trigonometric simplification

Simplifying trigonometric expressions involves using identities to make complex compositions appear simpler. In many cases, simplifying these expressions is necessary for problem-solving, particularly in integration or solving equations involving trigonometric functions.
Here's how trigonometric simplification works, taking our example step-by-step:

• Identify the pattern or identity that can be applied. For our exercise, we used the product-to-sum identity for simplifying \( 11 \sin \frac{x}{2} \cos \frac{x}{4} \).
• Substitute the angles into the identity. Calculate the required sums and differences. For instance, \( \frac{x}{2} + \frac{x}{4} = \frac{3x}{4} \) and \( \frac{x}{2} - \frac{x}{4} = \frac{x}{4} \).
• Simplify or reduce the expressions. We derived \( \frac{11}{2} ( \sin \frac{3x}{4} + \sin \frac{x}{4} ) \) as the final form.

This approach provides a way to visualize and transform trigonometric functions into their simplest forms, which is essential for further mathematical operations.

sine and cosine functions

Sine and cosine are foundational trigonometric functions that describe the relationships in right-angled triangles. They are also pivotal in modeling periodic phenomena, such as sound waves, and have profound applications across various scientific fields.
The sine and cosine functions are periodic, meaning they repeat their values in regular intervals. This characteristic makes them very predictable and extremely useful in solving trigonometric equations by transforming them into simpler forms.
When you see an expression like \( \sin \frac{x}{2} \) or \( \cos \frac{x}{4} \), understanding their periodic nature helps in predicting and validating simplifications. For example, their identities allow these functions to be manipulated into sums or differences through identities like the product-to-sum identity.

• Sine of an angle corresponds to the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.
• Cosine of an angle represents the adjacent side's length over the hypotenuse length.

Understanding these functions is crucial for mastering mathematical problem-solving in trigonometry.