Problem 61 - Algebra and Trigonometry

Question

\(61-66\) Write the sum as a product. $$ \sin 5 x+\sin 3 x $$

Step-by-Step Solution

Answer

The sum \(\sin 5x + \sin 3x\) can be expressed as the product \(2 \sin(4x) \cos(x)\).

Step 1: Identify the formula

To write the sum as a product of sines, we can use the sum-to-product identity for sine. The identity states that: \[ \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \] Here, identify \(A = 5x\) and \(B = 3x\).

Step 2: Apply the formula

Using the sum-to-product identity, substitute \(A = 5x\) and \(B = 3x\) into the formula: \[ \sin 5x + \sin 3x = 2 \sin \left( \frac{5x + 3x}{2} \right) \cos \left( \frac{5x - 3x}{2} \right) \]

Step 3: Simplify the expressions

Simplify the terms inside the sine and cosine functions: \[ \sin 5x + \sin 3x = 2 \sin \left( \frac{8x}{2} \right) \cos \left( \frac{2x}{2} \right) \] Which simplifies to: \[ \sin 5x + \sin 3x = 2 \sin (4x) \cos (x) \]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations involving trigonometric functions that hold true for all values of the involved variables. These identities are the key to simplifying or transforming complex trigonometric expressions.
They help in proving other mathematical properties or equations.
Some of the most commonly used trigonometric identities include:

• Pythagorean identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Angle sum and difference identities like \( \sin(A \pm B) \) and \( \cos(A \pm B) \).
• Sum-to-product and product-to-sum identities, such as the one used in this exercise: \( \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \).

These identities can transform a sum of sines or cosines into a product, making it easier for solving equations or simplifying expressions. It's important for students to understand and memorize these identities, as they can greatly simplify their work with trigonometric problems.

Sine Function

The sine function is one of the primary trigonometric functions, along with cosine and tangent. It is represented by \( \sin(\theta) \), where \( \theta \) is an angle. In the context of a unit circle, the sine of an angle is the y-coordinate of the point where the terminal side of the angle intersects the circle.
This property makes the sine function periodic with a period of \( 2\pi \).
Here's what you need to know about the sine function:

• It varies between -1 and 1 for all real numbers \( \theta \).
• The graph of the sine function is a smooth, continuous wave that repeats every \( 2\pi \) radians.
• Its key characteristic is its odd symmetry, meaning \( \sin(-\theta) = -\sin(\theta) \), which affects how equations are solved and manipulated.

Understanding these properties helps students use the sine function effectively in different mathematical situations, such as modeling waves or cyclic patterns. The sum-to-product identities are often used when combining multiple sine functions, as seen in the exercise where \( \sin 5x + \sin 3x \) was transformed using these identities.

Cosine Function

Like the sine function, the cosine function is fundamental in trigonometry, expressed as \( \cos(\theta) \), and it represents the x-coordinate of a point on the unit circle at angle \( \theta \). Its properties are very similar to the sine function, but with some differences.
Here's a closer look at its properties:

• The cosine function also oscillates between -1 and 1.
• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \), making it symmetric about the y-axis.
• Its period, like the sine function, is \( 2\pi \), resulting in a familiar wave pattern repeating every full circle.

These characteristics of the cosine function make it vital in problems involving symmetry and periods.
When combined with the sine function, as seen in trigonometric identities like the sum-to-product identity, it helps simplify expressions and solve equations.
In the exercise, we used cosine to express \( \cos(x) \) in the product form \( 2 \sin(4x) \cos(x) \), which simplified the trigonometric sum \( \sin 5x + \sin 3x \).