Problem 13 - Precalculus: Functions and Graphs

Question

Express as a product. $$\sin 3 t-\sin 7 t$$

Step-by-Step Solution

Answer

\( \sin 3t - \sin 7t = -2 \cos 5t \sin 2t \).

Step 1: Identify the Formula

The difference of two sines can be expressed as a product using the relevant trigonometric identity: \( \sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right) \). Identify that here \( A = 7t \) and \( B = 3t \).

Step 2: Calculate \( A + B \) and \( A - B \)

Compute the values \( A + B = 3t + 7t = 10t \) and \( A - B = 3t - 7t = -4t \).

Step 3: Apply the Formula

Substitute \( A + B = 10t \) and \( A - B = -4t \) into the difference of sines formula: \( \sin 3t - \sin 7t = 2 \cos \left( \frac{10t}{2} \right) \sin \left( \frac{-4t}{2} \right) \).

Step 4: Simplify the Expression

Simplify the expression: \( \cos \left( \frac{10t}{2} \right) = \cos 5t \) and \( \sin \left( \frac{-4t}{2} \right) = \sin (-2t) = -\sin 2t \). Thus, \( \sin 3t - \sin 7t = 2 \cos 5t (-\sin 2t) \).

Step 5: Express the Final Product

Combine the terms and simplify to express as a product: \( \sin 3t - \sin 7t = -2 \cos 5t \sin 2t \).

Key Concepts

Understanding the Sine Function

The sine function, symbolized as \( \sin \theta \), is the basis of our discussion today. In trigonometry, sine helps us understand angles and wave patterns.
When you look at the unit circle, sine measures the vertical height from the x-axis to the point on the circumference. For an angle \( \theta \), \( \sin \theta \) represents the y-coordinate of the corresponding point.
Key characteristics of the sine function:

• Periodic nature: The sine function repeats every \( 2\pi \) radians, meaning \( \sin(\theta + 2\pi) = \sin \theta \).
• Range: Its value always lies between -1 and 1 inclusive.
• Zero crossings: Occur at multiples of \( \pi \), such as \( 0, \pi, 2\pi, \ldots \).

When dealing with trigonometric identities, familiarity with the sine function is crucial. It's the foundation upon which more complex transformations and identities are built.

The Difference of Sines Identity

The identity for the difference of sines is a method derived from the properties of sine and cosine functions, which transforms a subtraction into a product. The identity is expressed as: \[\sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)\]This formula is especially useful when simplifying expressions or solving equations where the subtraction of two sine terms appears.
Here's how it works:

• First, calculate \( A + B \). Add the angles to find the average angle for the cosine function.
• Next, calculate \( A - B \). Subtract the second angle from the first to find the difference angle for the sine function.
• Finally, plug these results into the formula to transform the sine difference into a product of sine and cosine.

Applying this concept can make seemingly complex trigonometric problems manageable, by converting additions and subtractions into more tractable multiplications.

Using Product-to-Sum Formulas

Product-to-sum and sum-to-product formulas are powerful tools in trigonometry that provide a means to convert products of trigonometric functions into sums or differences, or vice versa. This is especially useful for simplifying expressions to a more manageable form. The exercise we analyzed applies the sum-to-product formula for sines, a method that transforms the difference of two sine functions. The general approach for using the product-to-sum formulas involves:

• Identifying the necessary identity to apply, based on the problem's structure.
• Substituting the given angles into the chosen formula.
• Simplifying the resulting expression to reach a cleaner, often multiplied form.

In our specific example, converting the sine difference into a product with \( -2 \cos 5t \sin 2t \) reflects the essence of this strategy. It highlights how combining trigonometric identities can lead to significant simplification and clarity in mathematical expression.