Problem 60 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall the Product-to-Sum Identity

The product-to-sum identity that relates the product of sine and cosine is \( 2 \sin A \cos B = \sin(A+B) + \sin(A-B) \). We'll use this identity to transform the given expression into a sum.

Step 2: Align the Expression with the Identity

In the expression \( 11 \sin \frac{x}{2} \cos \frac{x}{4} \), factor out the constant 11. Then use the identity by letting \( A = \frac{x}{2} \) and \( B = \frac{x}{4} \). Thus, \( 2 \sin A \cos B = \sin\left(\frac{x}{2} + \frac{x}{4}\right) + \sin\left(\frac{x}{2} - \frac{x}{4}\right) \).

Step 3: Simplify the Angles in the Identity

Calculate \( A + B = \frac{x}{2} + \frac{x}{4} = \frac{2x}{4} + \frac{x}{4} = \frac{3x}{4} \) and \( A - B = \frac{x}{2} - \frac{x}{4} = \frac{2x}{4} - \frac{x}{4} = \frac{x}{4} \).

Step 4: Apply the Entire Identity

Apply the identity to write \( 2 \sin \frac{x}{2} \cos \frac{x}{4} = \sin\left(\frac{3x}{4}\right) + \sin\left(\frac{x}{4}\right) \).

Step 5: Incorporate the Constant Factor 11

Since we factored out 11 initially, multiply back to obtain the final expression. Thus, \( 11 \sin \frac{x}{2} \cos \frac{x}{4} = 11 \cdot \frac{1}{2}(\sin\left(\frac{3x}{4}\right) + \sin\left(\frac{x}{4}\right)) = \frac{11}{2} (\sin\left(\frac{3x}{4}\right) + \sin\left(\frac{x}{4}\right)) \).

Key Concepts

Product-to-sum identities

The product-to-sum identities are a set of trigonometric formulas that help us rewrite the product of sine and cosine functions into a sum. This can often make it simpler to integrate or differentiate these expressions. For example, the identity for the product of

• sine of angle A and
• cosine of angle B

transforms the expression into a sum: \[ 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \].
This formula allows you to break down complex expressions into simpler sums of trigonometric functions. In the contextual example, by choosing \( A = \frac{x}{2} \) and \( B = \frac{x}{4} \), the process renders the expression in terms of the sum of two sine functions. Hence, these identities are crucial tools in algebraic manipulation within trigonometry and analysis. They are especially useful in converting problems into more solvable forms.

Sine function

The sine function, often denoted as \( \sin \theta \), is one of the fundamental functions in trigonometry. It is defined using the unit circle, but it also represents the y-coordinate of a point that is a certain angle away from the positive x-axis. The sine function is periodic with a period of \( 2\pi \).

• It starts at 0, peaks at 1, returns to 0, dips to -1, and returns to 0 again in one full cycle.
• The function is symmetrical and follows the properties: if \( \sin(\theta) = y \), then \( \sin(\pi - \theta) = y \) and \( \sin(-\theta) = -y \).

In applying the product-to-sum identities, the sine function helps facilitate rewriting products in the context of sums, particularly when transforming expressions like \( \sin(A + B) \) and \( \sin(A - B) \) from our original exercise. It is essential to recognize the sine function’s behavior and properties in these transformations as they dictate the nature of the resultant sums.

Cosine function

The cosine function, or \( \cos \theta \), is another core trigonometric function. Similar to the sine function, it is defined on the unit circle and represents the x-coordinate of a point a certain angle away from the positive x-axis. The cosine function is periodic with a period of \( 2\pi \).

• It begins at 1, reduces to 0, dips to -1, returns to 0, and back to 1 over one complete cycle.
• The cosine function has notable properties such as: \( \cos(\theta) = \cos(2\pi - \theta) \) and \( \cos(-\theta) = \cos(\theta) \).

In the context of product-to-sum identities, the cosine function plays a significant role in transitioning a multiplication of sine and cosine into a summation form. Understanding cosine's periodicity and symmetry supports working through trigonometric identities accurately. In the example of \( 11 \sin \frac{x}{2} \cos \frac{x}{4} \), recognizing the role of cosine in these transformations is key to deriving the resultant sum.