Problem 60 - Algebra and Trigonometry

Question

\(55-60\) Write the product as a sum. $$ 11 \sin \frac{x}{2} \cos \frac{x}{4} $$

Step-by-Step Solution

Answer

\( 11 \sin \frac{x}{2} \cos \frac{x}{4} = \frac{11}{2} [\sin \frac{3x}{4} + \sin \frac{x}{4}] \).

Step 1: Identify the Trigonometric Identity

The expression involves the product of sine and cosine functions. We can use the identity for the product of sine and cosine: \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \).

Step 2: Assign Variables

Assign \( A = \frac{x}{2} \) and \( B = \frac{x}{4} \) based on the given expression \( 11 \sin\frac{x}{2}\cos\frac{x}{4} \).

Step 3: Apply the Identity

Substitute \( A \) and \( B \) in the identity: \[ \sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{4}\right) = \frac{1}{2}\left[\sin\left(\frac{x}{2}+\frac{x}{4}\right) + \sin\left(\frac{x}{2}-\frac{x}{4}\right)\right] \].

Step 4: Simplify the Angles

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

Step 5: Write the Expression as a Sum

Substitute the simplified angles back into the identity: \[ \sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{4}\right) = \frac{1}{2}\left[\sin\left(\frac{3x}{4}\right) + \sin\left(\frac{x}{4}\right)\right] \].

Step 6: Apply the Scalar Multiplication

Multiply the entire expression by 11 (as it was the original coefficient): \[ 11\sin\frac{x}{2}\cos\frac{x}{4} = \frac{11}{2}\left[\sin\left(\frac{3x}{4}\right) + \sin\left(\frac{x}{4}\right)\right] \].

Key Concepts

Product-to-Sum Formulas

The product-to-sum formulas are a set of trigonometric identities that help transform the product of sine and cosine functions into a sum or difference. This transformation simplifies the process of solving and integrating trigonometric expressions.

• The basic idea is to convert products like \( \sin(A)\cos(B) \) into a more manageable sum form: \( \frac{1}{2}[\sin(A+B) + \sin(A-B)] \).
• This is particularly useful in calculus for integration and solving equations, as sums are often easier to work with than products.

In our example, we used this formula to break down the initial product expression \( 11 \sin \frac{x}{2} \cos \frac{x}{4} \) into a sum that is easier to analyze and compute. The product-to-sum identity lends itself well to various applications in both pure and applied mathematics.

Sine and Cosine Functions

Sine and cosine functions are fundamental in trigonometry and describe the relationship between the angles and sides of a right triangle. In the unit circle, these functions also represent the coordinates of a point.

• Side by side, they're known as periodic functions, with the sine function oscillating between -1 and 1.
• The cosine function also oscillates between -1 and 1 but is phase-shifted horizontally by 90 degrees compared to the sine.

In trigonometric identities, such as the product-to-sum formula, the sine and cosine functions interact beautifully to form new equations that help in simplifying complex expressions or solving equations. Thus, the transformation of a product of sine and cosine to sums can give us insights that might not have been immediately obvious.

Angle Addition and Subtraction

Understanding angle addition and subtraction is crucial for manipulating trigonometric expressions. These principles allow us to calculate the sine, cosine, and tangent of sums and differences of angles efficiently.

• The angle addition formula for sine is \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).
• Similarly, the subtraction formula is \( \sin(A - B) = \sin A \cos B - \cos A \sin B \).

When simplifying the expression \( \sin \left(\frac{x}{2} + \frac{x}{4}\right) \) and \( \sin \left(\frac{x}{2} - \frac{x}{4}\right) \), these rules enable us to correctly calculate the results.In the context of our task, these formulas allowed us to derive the simplified form of the expression, illustrating the elegance and utility of these trigonometric identities.