Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

From a product identity, we obtain $$ \cos \frac{x}{2} \cos \frac{x}{4}=\frac{1}{2}\left[\cos \left(\frac{3}{4} x\right)+\cos \left(\frac{1}{4} x\right)\right] $$

Step-by-Step Solution

Answer

Both sides of the equation match using the product-to-sum identity.

Step 1: Understand the Identities

In trigonometry, product-to-sum identities help us express products of trigonometric functions as sums or differences. One such identity states:\[\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]\]Here, we need to match this form to the given product \( \cos \frac{x}{2} \cos \frac{x}{4} \).

Step 2: Match Given Terms to Identity

In the identity \( \cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)] \), identify \( A \) and \( B \). In the given equation:- Let \( A = \frac{x}{2} \)- Let \( B = \frac{x}{4} \)

Step 3: Apply the Product-to-Sum Identity

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

Step 4: Simplify the Expression

Simplify each term inside the cosine functions:- \( \frac{x}{2} + \frac{x}{4} = \frac{3x}{4} \)- \( \frac{x}{2} - \frac{x}{4} = \frac{x}{4} \)Thus, the expression becomes:\[\cos \frac{x}{2} \cos \frac{x}{4} = \frac{1}{2}\left[\cos\left(\frac{3x}{4}\right) + \cos\left(\frac{x}{4}\right)\right]\]

Step 5: Verify the Solution

Check that the simplified expression is equivalent to the right-hand side of the original statement. Indeed it matches:\[\frac{1}{2}\left[\cos\left(\frac{3x}{4}\right) + \cos\left(\frac{x}{4}\right)\right]\]So, the identity holds true as given in the problem.

Key Concepts

Product-to-Sum Identities

Product-to-sum identities in trigonometry are essential tools that convert products of trigonometric functions into sums or differences. They simplify complex trigonometric expressions, making calculations easier.

• A common product-to-sum identity is for cosine: \( \cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)] \).
• These identities are particularly useful in integration and solving trigonometric equations.

Let's break this down: To express a product such as \( \cos \frac{x}{2} \cos \frac{x}{4} \) as a sum, identify values for \( A \) and \( B \) in terms of \( x \):* Set \( A = \frac{x}{2} \).* Set \( B = \frac{x}{4} \).Plugging these into the identity transforms the product into a sum, simplifying further manipulation of the expression. This technique is powerful for handling trigonometric computations in calculus and physics problems.

Cosine Function

The cosine function, denoted as \( \cos \), is one of the primary trigonometric functions. It is defined on the unit circle as the x-coordinate of a point where an angle's terminal side intersects the circle.

• The cosine function ranges from -1 to 1.
• It has a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians.
• Key points include \( \cos(0) = 1 \), \( \cos(\pi/2) = 0 \), and \( \cos(\pi) = -1 \).

When using the cosine function in product-to-sum identities, it's crucial to understand its properties:- **Symmetry**: \( \cos(-x) = \cos(x) \), meaning it's an even function.- **Shifts**: Understanding phase shifts help in graphing transformations and solving trigonometric problems.Recognizing these attributes allows for accurate application of trigonometric identities, such as converting products into sums. This is key when simplifying expressions like \( \cos \frac{x}{2} \cos \frac{x}{4} \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. These identities simplify and solve expressions and equations. Some fundamental categories of trigonometric identities include:- **Pythagorean Identities**: Expressions that relate \( \sin \), \( \cos \), and \( \tan \).- **Angle Sum and Difference Identities**: For finding trigonometric function values of sums or differences of angles.- **Double Angle and Half Angle Identities**: Useful in calculus and analysis for simplifying expressions and solving integrals.In the exercise, the focus is on product-to-sum identities, a type that helps translate products of cosines and sines into sums or differences, aiding in easier manipulation or integration.These identities are essential for problem-solving across various math and engineering fields. They allow for significant simplification when encountering multiple trigonometric functions in a product form.