Problem 7 - Precalculus Essentials

Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working \(1-4\) in the Concept and Vocabulary Check. Use the appropriate formula to express each product as a sum or difference. $$\cos \frac{3 x}{2} \sin \frac{x}{2}$$

Step-by-Step Solution

Answer

The product \(\cos \frac{3 x}{2} \sin \frac{x}{2}\) can be expressed as a sum or difference: \(1/2[sin(2x) - sin(x)]\).

Step 1: Identify the product-to-sum identity to use

From the identities of product-to-sum formulas, we can see that our given expression falls under the form of \(cos(A)sin(B)\). The appropriate identity to use here is: \[cos(A)sin(B) = 1/2[sin(A + B) - sin(A - B)]\]

Step 2: Substitute the values of A and B

Here, we can see that \(A = \frac{3x}{2}\) and \(B = \frac{x}{2}\). Plug these values into the identity from step 1.

Step 3: Calculate the sum and difference

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

Step 4: Substitute into the product-to-sum formula

Now, we substitute the results of sum and difference from step 3 into the formula from step 1: \[cos(\frac{3x}{2})sin(\frac{x}{2}) = 1/2[sin(2x) - sin(x)]\]

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are a powerful tool in trigonometry. They provide a way to express products of trigonometric functions as sums or differences, which can simplify analysis and computation. This particular type of formula helps us change the product of functions like cosine and sine into a more manageable form. For example, the product-to-sum formula that relates a product of cosine and sine is:

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

This identity works by converting the product of \( \cos(A) \) and \( \sin(B) \) into the sum of two sine functions, thereby making it easier to handle or integrate into larger equations. When using these formulas, it's crucial to recognize the structure you are working with (e.g., cos-sin, sin-sin, cos-cos) and choose the appropriate formula accordingly. Doing so allows for efficient problem-solving and often leads to more straightforward interpretations of trigonometric expressions.

Cosine and Sine Functions

Cosine and sine functions are fundamental in trigonometry. They are periodic functions that are highly useful for modeling wave-like phenomena. - **Cosine Function:** The cosine function, denoted \( \cos(x) \), tells us how much of a wave is above or below the baseline, depending on the angle.- **Sine Function:** The sine function, denoted \( \sin(x) \), represents how far along the baseline the wave has moved. Both functions have a range of -1 to 1 and are defined based on the unit circle. On this circle, the value of these functions is related to the position of a point decreed by a given angle. This makes them extremely versatile in expressing angles and rotations. Together, cosine and sine enable transformations of trigonometric expressions into different formats, such as changing products to sums, which can make complex calculations more accessible.

Trigonometric Expressions

Trigonometric expressions are any mathematical expressions that involve trigonometric functions like sine, cosine, tangent, and others. Solving problems involving these expressions often involves transforming them using identities and formulas, such as product-to-sum formulas. These expressions can be simplified by recognizing patterns or using standard transformation techniques. For instance, in a given problem like \( \cos \left( \frac{3x}{2} \right) \sin \left( \frac{x}{2} \right) \), trigonometric identities help us rewrite this product into a more convenient form using the product-to-sum identity, resulting in: \[ \frac{1}{2}[\sin(2x) - \sin(x)] \] Understanding the structure of trigonometric expressions helps us choose the right identity to achieve simplification, thus leading to easier analysis or further computations. As students progress, becoming adept at identifying these expressions and knowing which formulas to apply is invaluable for effective problem-solving in trigonometry.