Problem 11 - Precalculus with Calculus Previews

Question

Use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ 2 \sin \left(x+\frac{\pi}{4}\right) \sin \left(x-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-\cos(2x)\).

Step 1: Recall the Product-to-Sum Formula

The product-to-sum formulas allow us to express the product of sines as a sum: \[ 2 \sin A \sin B = \cos(A-B) - \cos(A+B) \]. Identifying the angles here, we have \( A = x + \frac{\pi}{4} \) and \( B = x - \frac{\pi}{4} \).

Step 2: Apply the Product-to-Sum Formula

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

Step 3: Simplify the Expressions

Calculate \( A - B \) and \( A + B \) to simplify:\[ (x + \frac{\pi}{4}) - (x - \frac{\pi}{4}) = \frac{\pi}{2} \]\[ (x + \frac{\pi}{4}) + (x - \frac{\pi}{4}) = 2x \].Substitute these into the equation:\[ \cos\left(\frac{\pi}{2}\right) - \cos(2x) \].

Step 4: Calculate Cosines

Evaluate the cosine expressions: \( \cos\left(\frac{\pi}{2}\right) = 0 \), so the expression becomes:\[ 0 - \cos(2x) \].

Step 5: Conclude the Result

The original expression \( 2 \sin\left(x + \frac{\pi}{4}\right) \sin\left(x - \frac{\pi}{4}\right) \) simplifies to \( -\cos(2x) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. They are used extensively in simplifying expressions and solving equations. Some common trigonometric identities include:

• Pythagorean identities, such as \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle sum and difference identities, for example, \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
• Double angle identities like \( \sin(2x) = 2 \sin x \cos x \)

Trigonometric identities help in transforming products into sums or differences. This transformation is valuable in simplifying complicated trigonometric expressions. Recognizing these identities is key to effectively tackling trigonometric problems.

Sum of Cosines

In trigonometry, there are various situations where we transition from a product expression to a sum, particularly involving the cosine function. The sum of cosines is a result achieved through the product-to-sum formulas. These formulas make calculations easier by converting products into more manageable sums or differences of trigonometric functions.Using the formula \( \cos(A - B) - \cos(A + B) \), we can express the product of two sine functions as a sum of cosines. This specific transformation is especially useful in applications such as signal processing and physics, where sinusoidal expressions frequently appear. Understanding how cosine sums function can enhance one's ability to work with periodic functions and solve real-world problems.

Sum of Sines

The sum of sines can also be derived from the product-to-sum formulas, similar to the sum of cosines. By turning products of trig functions into sums, these identities allow for simplification of expressions or equations, particularly those appearing in calculus, engineering, and physics.The product-to-sum formula for sine works as follows:

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

These formulas can showcase how combinations of sines and cosines behave under addition or subtraction, equipping learners with the tools to simplify complex trigonometric problems.