Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the product-to-sum formulas to write the product as a sum or difference. $$5 \sin \theta \sin 3 \theta$$

Step-by-Step Solution

Answer

\(\frac{5}{2}(\cos(-2\theta) - \cos(4\theta))\)

Step 1: Identify Variables

Set \(A = \theta\), and \(B = 3 \theta\).

Step 2: Apply Formula

Apply the product-to-sum formula using the identified variables. According to the formula \(2\sin(A)\sin(B) = \cos(A-B) - \cos(A+B)\), substitute \(A\) and \(B\). But notice the coefficient on the left-hand side of the identity is 2, and the coefficient given in the problem is 5 which leads us to multiply the resultant answer by \(\frac{5}{2}\).

Step 3: Substitute the Terms

Substituting \(A\) and \(B\) into the right-hand side of the identity leads to: \(\cos(\theta - 3\theta) - \cos(\theta + 3\theta)\). Simplify the terms inside the cosine functions to get: \(\cos(-2\theta) - \cos(4\theta)\). Multiplying this by \(\frac{5}{2}\) gives the final expression.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved, provided they are within the domain of the trigonometric functions. They are essential tools in trigonometry and calculus and help us simplify complex expressions or solve equations more efficiently.
In this exercise, we specifically deal with the product-to-sum formulas. These formulas convert products of sines and cosines into simpler expressions that involve sums or differences of cosines. Here is the formula we used:

• For two sine functions: \(2 \sin A \sin B = \cos(A - B) - \cos(A + B)\)

Understanding these identities not only helps solve various trigonometric problems but also provides insight into the relationships between different trigonometric functions. Recognizing which identity can simplify a problem relies heavily on practice and familiarity with all the main trigonometric identities.
This exercise is a perfect illustration of why understanding identities like the product-to-sum formula is so valuable.

Cosine Function

The cosine function, denoted as \(\cos\), is one of the primary trigonometric functions. It is particularly useful in this exercise because of its role in the product-to-sum formula.
The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. On the unit circle, it represents the projection of the arc onto the horizontal axis.

• The cosine function is even, meaning \(\cos(-x) = \cos(x)\). This characteristic was useful when simplifying \(\cos(-2\theta)\) to \(\cos(2\theta)\).

In our solution, recognizing the cosine terms \(\cos(\theta - 3\theta)\) and \(\cos(\theta + 3\theta)\) was crucial for applying the product-to-sum formula. By substituting into the formula and simplifying these cosine terms, we reach the final expression. The cosine function thus played a pivotal role in transforming the original trigonometric product into an addition/subtraction form.

Sine Function

The sine function, represented by \(\sin\), is another fundamental trigonometric function. In this exercise, the function is part of the initial expression we are dealing with: \(5 \sin \theta \sin 3\theta\).
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse. On the unit circle, the sine function corresponds to the vertical distance from the point on the circle to the x-axis.

• The sine function is odd, which means it has the property \(\sin(-x) = -\sin(x)\).

In this exercise, we leveraged the fact that the sine function appears in the product we needed to transform. By utilizing trigonometric identities like the product-to-sum formulas, expressions involving products of sine functions can often be simplified into sums or differences of cosines. This transformation is particularly useful in both integration and solving equations. Understanding the sine function's properties and its connection to other trigonometric functions is essential for applying these identities correctly.