Problem 39 - Algebra and Trigonometry

Question

(a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ H(x)=2 \sin (3 x) \cos (x) $$

Step-by-Step Solution

Answer

Using the Product-to-Sum formula, \( H(x) = 2 \sin(3x) \cos(x) \) simplifies to \( \sin(4x) + \sin(2x) \). Graph the resultant function on the interval \[0, 2 \pi \].

Step 1: Identify the Product-to-Sum Formula

Recognize the given product \( \sin(A) \cos(B) \) can be expressed as a sum using the product-to-sum formula: \[ \sin(A) \cos(B) = \frac{1}{2} \[ \sin(A+B) + \sin(A-B) \] \]

Step 2: Apply the Product-to-Sum Formula

In the equation \(H(x) = 2 \sin(3x) \cos(x)\), identify \A = 3x\ and \B = x\. Substitute these values into the product-to-sum formula: \[ \sin(3x) \cos(x) = \frac{1}{2} \[ \sin(4x) + \sin(2x) \] \]

Step 3: Simplify the Expression

Multiply the entire equation by 2 to account for the original coefficient: \[ H(x) = 2 \left(\frac{1}{2} \left( \sin(4x) + \sin(2x) \right) \right) = \sin(4x) + \sin(2x) \]

Step 4: Graphing the Function

To graph \( H(x) = \sin(4x) + \sin(2x) \) on the interval \ [0, \2 \pi] \, construct a table of values for \x\ and calculate the corresponding values of \H(x)\ for \[ 0, \ \frac{\pi}{6}, \ \frac{\pi}{4}, \ \frac{\pi}{3}, \ \frac{\pi}{2}, \ \frac{2\pi}{3}, \ \pi, \ \frac{4\pi}{3}, \ \frac{3\pi}{2}, \ \frac{5\pi}{3}, \2\pi \]. Plot these points and connect them to form the graph.

Key Concepts

trigonometric identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values within their domain. They are crucial in simplifying expressions and solving equations. One important identity is the Product-to-Sum Formula, which allows us to express products of sine and cosine functions as sums or differences of sines. This identity can be written as: \[ \text{sin}(A) \text{cos}(B) = \frac{1}{2}[\text{sin}(A+B) + \text{sin}(A-B)] \] Using this identity helps in breaking down complex trigonometric expressions into simpler ones. For example, in the given problem, we start with the function \(H(x) = 2 \text{sin}(3x) \text{cos}(x)\). By applying the Product-to-Sum formula, we transform it into a sum of sines, making it easier to handle and graph.

function graphing

Graphing functions is a fundamental skill in mathematics, providing visual insight into the behavior of the function over a specified interval. To graph the function \(H(x) = \text{sin}(4x) + \text{sin}(2x)\) on the interval \([0, 2\text{π}]\), follow these steps:

• Construct a table of values. Choose specific values within the interval, such as \(0, \frac{\text{π}}{6}, \frac{\text{π}}{4}, \frac{\text{π}}{3}, \frac{\text{π}}{2}, \frac{2\text{π}}{3}, \text{π}, \frac{4\text{π}}{3}, \frac{3\text{π}}{2}, \frac{5\text{π}}{3}, \text{2π}\).
• Calculate the corresponding \(H(x)\) values using the simplified expression \(\text{sin}(4x) + \text{sin}(2x)\).
• Plot these points on the coordinate plane.
• Connect the points smoothly to reveal the shape of the graph.

This method of plotting points and connecting them visually represents the behavior of trigonometric functions, showing their periodic nature and how they combine in different ways.

trigonometric functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics. The sine function, \(\text{sin}(x)\), and the cosine function, \(\text{cos}(x)\), are periodic with a period of \(\text{2π}\). They describe the relationship between the angles of a right triangle and the lengths of its sides. Key properties of sine and cosine functions include:

• Their values range between -1 and 1.
• They are periodic with a period of \(\text{2π}\).
• The graphs of these functions are smooth and continuous, showing a repetitive wave-like pattern.

In this exercise, the given function \(H(x)\) involves both \(\text{sin}(4x)\) and \(\text{sin}(2x)\). Here, the coefficients of x determine the frequency of the sine waves. For \(\text{sin}(4x)\), the frequency is four times that of \(\text{sin}(x)\), leading to more oscillations per interval. By understanding these key aspects of trigonometric functions, you can better interpret and graph expressions involving them, like the given problem. This will aid in visualizing their combined effects and periodicity.