Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the product-to-sum formulas to write the product as a sum or difference. $$4 \cos \frac{\pi}{3} \sin \frac{5 \pi}{6}$$

Step-by-Step Solution

Answer

The product can be written as a sum: \(4 \cos \frac{\pi}{3} \sin \frac{5 \pi}{6} = -\frac{1}{2} - 1 = -\frac{3}{2}\)

Step 1: Apply the Product-to-Sum Formula

First, divide the term by 2, the constant in the product-to-sum formula, which leaves \(2 \cos \frac{\pi}{3} \sin \frac{5 \pi}{6}\). Then, use the product-to-sum formula, replacing the product of the cos and sin terms with \(\sin(a + b) + \sin(a - b)\). This gives \(\sin \left(\frac{\pi}{3} + \frac{5 \pi}{6}\right) + \sin \left(\frac{\pi}{3} - \frac{5 \pi}{6}\right)\).

Step 2: Simplify Inside the Sine Functions

For the sum inside the first sine function, \(\frac{\pi}{3} + \frac{5 \pi}{6} = \frac{2 \pi}{6} + \frac{5 \pi}{6} = \frac{7 \pi}{6}\). For the difference inside the second sine function, \(\frac{\pi}{3} - \frac{5 \pi}{6} = \frac{2 \pi}{6} - \frac{5 \pi}{6} = -\frac{3 \pi}{6}\), which simplifies to \(-\frac{\pi}{2}\).

Step 3: Evaluate the Sine Functions

Evaluate the sine functions: \(\sin (\frac{7 \pi}{6}) = -\frac{1}{2}\) and \(\sin (-\frac{\pi}{2}) = -1\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential mathematical tools that relate various trigonometric functions to each other. They help in simplifying and transforming expressions involving trigonometric terms. There are several basic identities, such as the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), which are crucial for more complex equations.

In trigonometry, the product-to-sum formulas play an important role as well. These formulas allow us to convert products of sine and cosine into sums, which can simplify complicated expressions or make integration easier. For instance, one product-to-sum formula is:

• \( \cos(a) \sin(b) = \frac{1}{2} \left[ \sin(a+b) - \sin(a-b) \right] \).

These identities are used in various contexts, including solving integrals and analyzing wave patterns, making them fundamental in both academic and practical applications.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using mathematical techniques to transform complex trigonometric formulas into simpler ones. By doing this, we can make the computations simpler and more manageable. The process often involves applying trigonometric identities or formulas, such as the product-to-sum or sum-to-product formulas.

In the given exercise, applying the product-to-sum formula drastically simplifies the expression by replacing the product of \( \cos \frac{\pi}{3} \sin \frac{5 \pi}{6} \) into a sum of sines:

• First, factor out the constant 4 to make use of \( \frac{1}{2} \)
• Then, the Product-to-Sum formula is used: \( \sin \left(\frac{\pi}{3} + \frac{5\pi}{6}\right) + \sin \left(\frac{\pi}{3} - \frac{5\pi}{6}\right) \).

By breaking the product into a sum, you simplify the evaluation of the expression since evaluating single sin functions is easier than products of functions.

Sine and Cosine Functions

Sine and cosine functions are two of the most fundamental functions in trigonometry. They are periodic, meaning they repeat values in regular intervals, and they are crucial in describing oscillatory behavior such as sound waves, light waves, and circular motion.

The cosine function, \( \cos(\theta) \), represents the adjacent side over the hypotenuse in a right triangle, while the sine function, \( \sin(\theta) \), represents the opposite side over the hypotenuse. These functions are integral in calculating angles and distances in various scientific fields.

In our exercise, understanding how to manipulate these functions using trigonometric identities is key. For example, evaluating \( \sin \left(\frac{7\pi}{6}\right) \) and \( \sin(-\frac{\pi}{2}) \) involves understanding these functions' periodic behavior and using it to simplify or interpret expressions intuitively. The sine of \( \frac{7\pi}{6} \) results in \(-\frac{1}{2}\), and sine of \(-\frac{\pi}{2}\) is \(-1\), showing how they can be used effectively to solve trigonometric problems.