Problem 2 - 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. $$ \sin \frac{3 t}{2} \cos \frac{t}{2} $$

Step-by-Step Solution

Answer

\( \sin \frac{3t}{2} \cos \frac{t}{2} = \frac{1}{2} \sin(2t) + \frac{1}{2} \sin(t) \).

Step 1: Identify the Product-to-Sum Formula

The product-to-sum formulas allow us to express the product of sine and cosine as a sum. The specific formula needed here is: \( \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \).

Step 2: Plug in Values for A and B

For the term \( \sin \frac{3t}{2} \cos \frac{t}{2} \), we identify \( A = \frac{3t}{2} \) and \( B = \frac{t}{2} \). Substitute these into our formula: \[ \frac{1}{2} [\sin(\frac{3t}{2} + \frac{t}{2}) + \sin(\frac{3t}{2} - \frac{t}{2})] \].

Step 3: Simplify the Expressions Inside the Sines

Calculate \( A + B \) and \( A - B \): \( \frac{3t}{2} + \frac{t}{2} = 2t \) and \( \frac{3t}{2} - \frac{t}{2} = t \). Substitute these back into our expression to get: \[ \frac{1}{2} [\sin(2t) + \sin(t)] \].

Step 4: Write Down the Final Answer

The product \( \sin \frac{3t}{2} \cos \frac{t}{2} \) is equivalent to the sum of sines: \[ \frac{1}{2} \sin(2t) + \frac{1}{2} \sin(t) \].

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that involve trigonometric functions, which relate the angles and sides of triangles. These identities are vital in mathematics because they allow us to simplify and manipulate complex trigonometric expressions. One of the core reasons for using trigonometric identities is to transform expressions into a more workable or understandable form. This becomes especially useful in calculus, physics, and engineering.

For example, product-to-sum formulas, a subset of trigonometric identities, transform products of trigonometric functions into sums or differences. This particular transformation simplifies problems by converting multiplicative relationships, which are often more complex to manipulate, into additive ones, which are generally easier to handle. Understanding these formulas can be a significant advantage when solving trigonometric equations or integrals.

To visualize, consider the product-to-sum identity:

• Product-to-sum formula: \( \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \)

The transformation provided by this identity allows us to solve problems where multiplication occurs between sine and cosine functions with ease.

Sine and Cosine Functions

The sine and cosine functions are fundamental in describing periodic phenomena. In their basic forms, these functions represent the relationships between the angles and sides in a right triangle, or as the coordinates on a unit circle. Importantly, both functions are periodic with a period of \(2\pi \), repeating their values at regular intervals.

Sine and cosine functions have several key features:

• Amplitude: This is the height of the wave from the center line to the peak.
• Period: This is the distance over which the wave pattern repeats.
• Phase shift and vertical shift: These allow the wave to move left, right, up, or down along the graph.

When you multiply sine and cosine functions, you encounter products of waves, as in the exercise with \( \sin \frac{3 t}{2} \cos \frac{t}{2} \). Using product-to-sum identities simplifies these products into sums of sines or cosines, which is much easier to interpret and work with.

Mathematical Simplification

Mathematical simplification is a crucial skill that helps solve complex problems by reducing them to more manageable forms. This process involves applying mathematical rules and identities to transform expressions for easier computation or deeper understanding. Simplification lets you see the core behavior of a function or equation with greater clarity.

In the realm of trigonometry, simplifications often involve using identities like the sum-to-product, product-to-sum, and double angle formulas. When you look at an expression such as \( \sin \frac{3t}{2} \cos \frac{t}{2} \), simplifying it using a product-to-sum formula gives us:

• \( \frac{1}{2} [\sin(2t) + \sin(t)] \)

This approach reduces a product into a sum, making it easier to integrate, differentiate, or use in further calculations. By mastering mathematical simplification, you can efficiently navigate complex trigonometric functions and tackle a broader range of mathematical challenges.