Problem 17 - Precalculus Essentials

Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working \(5-8\) in the Concept and Vocabulary Check. Express each sum or difference as a product. If possible, find this product's exact value. $$\cos \frac{3 x}{2}+\cos \frac{x}{2}$$

Step-by-Step Solution

Answer

This given sum can be expressed as a product i.e., \[2\cos(2x) \cos(x)\]. The exact value will depend on the specific value of x. If x can be represented in terms of multiples of \(\pi\), further simplification may be possible to find an exact value.

Step 1: Recognize the Requirement

Recognize what the problem is asking for. We need to express the given sum \(\cos \frac{3 x}{2}+\cos \frac{x}{2}\) as a product using sum-to-product identities.

Step 2: Apply Sum-to-Product Formula

The sum-to-product identities are as follows: \[\cos(a) + \cos(b) = 2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)\] \[\cos(a) - \cos(b) = -2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)\] These identities can be used to convert the sum or difference of two cosine or sine functions into the product of sine or cosine functions. Apply the suitable sum-to-product identity formula to our given statement.

Step 3: Substitute the Values

Let \(a = \frac{3x}{2}\) and \(b = \frac{x}{2}\). After substituting these values into our chosen identity, we obtain: \[2\cos\left(\frac{\frac{3x}{2} + \frac{x}{2}}{2}\right)\cos\left(\frac{\frac{3x}{2} - \frac{x}{2}}{2}\right)\]. Reducing and simplifying this expression yields our product.

Step 4: Calculate the Exact Value

If possible, compute the exact value of the product. In this case, it would depend on the value of x. If x is known, substitute it into the product and compute the value.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all permitted variable values, involving trigonometric functions. A notable category of these identities is the sum-to-product formulas. These are particularly useful when we need to transform an addition or subtraction of two trigonometric expressions into a product. This transformation can greatly simplify expressions and solve equations effectively. For example, in the problem presented, we use the identity:

• \(\cos(a) + \cos(b) = 2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)\)

The key advantage of such identities is how they can help us manipulate and simplify trigonometric expressions to make them more manageable. Understanding and applying these identities is a critical skill in precalculus, unlocking more advanced problem-solving methods.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is vital in various mathematical situations. It describes the x-coordinate of a point on the unit circle as it relates to the angle formed with respect to the positive x-axis. Cosine function values range between -1 and 1, and the function shows periodic behavior with a period of

• \(2\pi\).

In our exercise, the expression \(\cos \frac{3x}{2} + \cos \frac{x}{2}\) is tackled using trigonometric identities. This sum-to-product transformation helps transform the sum of two cosine expressions into a product, making subsequent calculations or applications more straightforward. By exploring these transformations, you will gain deeper insight into how the cosine function can be analyzed and manipulated to solve complex problems.

Precalculus Problem-Solving

Precalculus problem-solving often involves the application of various mathematical identities to simplify and solve equations. One common scenario is transforming trigonometric sums or differences, like our cosine example, into a more manageable form. By expressing trigonometric expressions as products, the complexities of further calculations are reduced significantly. This makes it easier to find exact values or parametrically model the behavior of trigonometric functions.

Successfully solving these problems involves recognizing which identities to apply—such as sum-to-product or vice versa—and skillfully manipulating the expressions. Mastery of these techniques will enhance your mathematical problem-solving arsenal, providing a solid foundation for more advanced studies ranging from calculus to physics and engineering.