Problem 10 - Precalculus: Functions and Graphs

Question

Express as a product. $$\sin 4 \theta-\sin 8 \theta$$

Step-by-Step Solution

Answer

\(\sin 4\theta - \sin 8\theta = 2 \cos 6\theta \sin 2\theta\).

Step 1: Identify Trigonometric Identity

To express the difference of two sine functions as a product, we can use the sum-to-product identities. Specifically, we use the identity for the difference of sines: \[\sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)\]where \(A = 8\theta\) and \(B = 4\theta\).

Step 2: Calculate \(A + B\)

Calculate the sum of \(A\) and \(B\):\[A + B = 8\theta + 4\theta = 12\theta.\]

Step 3: Calculate \(A - B\)

Calculate the difference of \(A\) and \(B\):\[A - B = 8\theta - 4\theta = 4\theta.\]

Step 4: Apply the Identity

Substitute \(A + B\) and \(A - B\) into the formula:\[\sin 4\theta - \sin 8\theta = 2 \cos \left(\frac{12\theta}{2}\right) \sin \left(\frac{4\theta}{2}\right).\]

Step 5: Simplify the Expression

Simplify both terms:\[\cos \left(\frac{12\theta}{2}\right) = \cos 6\theta, \quad \sin \left(\frac{4\theta}{2}\right) = \sin 2\theta.\]Thus, the expression becomes:\[2 \cos 6\theta \sin 2\theta.\]

Key Concepts

Sum-to-Product Identities

Trigonometric identities provide us with useful tools to simplify and manipulate expressions involving trigonometric functions. Among these, the sum-to-product identities allow us to rewrite the sum or difference of two trigonometric functions into a product. This transformation can make integration and differentiation processes easier.

The crucial identity used here is for the difference of sines:

• \( \sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right) \)

This identity can be used in problems where two sine functions are subtracted, turning them into a single product term. This is particularly powerful in solving and simplifying expressions, like in integrals where product forms are often easier to handle.

Understanding how and when to apply these identities is key to solving trigonometric problems effectively.

Difference of Sines

The difference of sines is a specific case used in trigonometric identities to convert a subtraction of two sine functions into a product. This conversion is beneficial as it often simplifies the problem at hand, especially when dealing with trigonometric equations or calculus problems.

For the identity \( \sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right) \), it’s important to first identify what your \(A\) and \(B\) are.

In the exercise \(\sin 4\theta - \sin 8\theta\), you identify \(A = 8\theta\) and \(B = 4\theta\). Once identified, substitute these into the identity, but first calculate \(A+B\) and \(A-B\) to find the angles used in the cosine and sine functions in the identity.

This strategy significantly streamlines solving more complex trigonometric problems by reducing the number of terms you're dealing with.

Trigonometric Simplification

Trigonometric simplification is a fundamental skill in mathematics that involves reducing expressions to simpler or more standard forms. This is particularly useful in calculus, algebra, and geometry.

In the example of \(\sin 4\theta - \sin 8\theta\), once the sum-to-product identity has been applied, you must simplify the result: \(2 \cos 6\theta \sin 2\theta\).

The simplification involves:

• Calculating \(\cos(6\theta)\) instead of dealing with the more complicated \(\cos\left(\frac{12\theta}{2}\right)\).
• Reducing \(\sin(4\theta/2)\) to \(\sin(2\theta)\).

Such simplifications can make further calculations more straightforward, whether you are graphing functions, calculating derivatives, or performing integrals.
A firm understanding of these processes helps in passing exams and solving real-world applications that involve trigonometric functions.