Problem 10 - Precalculus with Calculus Previews

Question

Use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ \sin \pi \theta \cos 7 \pi \theta $$

Step-by-Step Solution

Answer

\( \frac{1}{2}(\sin(8\pi \theta) - \sin(6\pi \theta)) \)

Step 1: Identify the Appropriate Product-to-Sum Formula

The given expression is \( \sin A \cos B \). Recall that the product-to-sum formula for \( \sin A \cos B \) is:\[ \sin A \cos B = \frac{1}{2}(\sin(A+B) + \sin(A-B)) \]

Step 2: Substitute Values into the Formula

In our expression, \( A = \pi \theta \) and \( B = 7 \pi \theta \). Substitute these into the formula:\[ \sin \pi \theta \cos 7 \pi \theta = \frac{1}{2}(\sin(\pi \theta + 7 \pi \theta) + \sin(\pi \theta - 7 \pi \theta)) \]

Step 3: Simplify the Expressions Inside the Sine Functions

Calculate \( A+B \) and \( A-B \):- \( A+B = \pi \theta + 7\pi \theta = 8\pi \theta \)- \( A-B = \pi \theta - 7\pi \theta = -6\pi \theta \).Thus, the expression becomes:\[ \frac{1}{2}(\sin(8\pi \theta) + \sin(-6\pi \theta)) \]

Step 4: Simplify Using Properties of Sine Function

Using the identity \( \sin(-x) = -\sin(x) \), you can simplify \( \sin(-6\pi \theta) \) to \(-\sin(6\pi \theta) \).Our final expression is:\[ \frac{1}{2}(\sin(8\pi \theta) - \sin(6\pi \theta)) \]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved trigonometric functions, making them a powerful tool in simplifying complex trigonometric expressions. In this context, the product-to-sum formulas are a group of trigonometric identities that help us express products of trigonometric functions as sums or differences. Coming from Euler's formula, these identities simplify integration and complex calculations in trigonometry.
\( \sin A \cos B = \frac{1}{2}(\sin(A+B) + \sin(A-B)) \) is an example of such an identity. With it, we can break down the product of sine and cosine into simpler sine terms. This can make solving problems involving these functions a lot easier. Understanding these identities helps you manipulate and solve problems much faster and is key in areas such as calculus, physics, and engineering.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry. They describe the y and x coordinates of a point on a circle of radius one, known as the unit circle, given an angle from the positive x-axis.
Sine and cosine functions are periodic, oscillating between -1 and 1, with a period of \(2\pi\). They are essential for modeling waves and other oscillatory behaviors.
Here are some crucial properties of sine and cosine functions:

• Symmetry: Sine is an odd function, \( \sin(-x) = -\sin(x) \), while cosine is an even function, \( \cos(-x) = \cos(x) \).
• Range: Both sine and cosine have a range of [-1, 1].
• Fundamental Period: Both functions repeat every \(2\pi\).

Knowing these properties is essential when applying product-to-sum formulas and transforming expressions like \(\sin \pi \theta \cos 7 \pi \theta \) into simpler forms.

Angle Transformation

Angle transformation involves changing trigonometric expressions using identities that alter angle measures. This is crucial in trigonometry when moving terms around to line up a suitable identity or when simplifying expressions.
In our solution, the angles \(A = \pi \theta \) and \(B = 7\pi \theta \) are transformed through addition and subtraction:

• Addition: \(A + B = 8\pi \theta\)
• Subtraction: \(A - B = -6\pi \theta\)

These kinds of transformations are vital in working with trigonometric equations. Using properties like \( \sin(-x) = -\sin(x) \), we can further simplify results. This demonstrates how angle transformations allow us to derive elegant solutions to complex products—by reformulating them into sums or differences. Understanding and applying angle transformations are useful in tackling trigonometric equations effectively.