Problem 85 - Precalculus with Limits

Question

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( \sin 5 \theta \sin 3 \theta \)

Step-by-Step Solution

Answer

The product \( \sin 5 \theta \sin 3 \theta \) can be expressed as a difference \( (1/2) [\cos(2 \theta) - \cos(8 \theta)] \).

Step 1: Identify the product-to-sum formula for sine functions

The product-to-sum formula for sine functions is given by: \( \sin A \sin B = (1/2) [\cos(A - B) - \cos(A + B)] \). In this problem, A is 5 theta and B is 3 theta.

Step 2: Substitute A and B into the formula

Let's substitute A as 5 theta and B as 3 theta into the formula: \( \sin(5 \theta) \sin(3 \theta) = (1/2) [\cos(5 \theta - 3 \theta) - \cos(5 \theta + 3 \theta)] \)

Step 3: Simplify the expression

After substitution, simplify the expression: \( \sin(5 \theta) \sin(3 \theta) = (1/2) [\cos(2 \theta) - \cos(8 \theta)] \). This is the expression of the product as a difference, as required.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable where both sides of the equation are defined. These identities are crucial in the field of mathematics, especially in geometry and calculus, as they allow the simplification and transformation of trigonometric expressions, which can make calculations more manageable and can also provide insight into the properties of angles and triangles.

The product-to-sum formulas are a set of trigonometric identities that express the product of two sine or cosine functions as the sum or difference of two cosines or sines, respectively. They are particularly useful in integrations, solving trigonometric equations, and in Fourier series. For example, the product-to-sum formula for two sine functions, used in the given exercise, is expressed as:
\( \sin A \sin B = \frac{1}{2} \left[\cos(A - B) - \cos(A + B)\right] \)
By using this identity, the multiplication of sine functions is translated into a more straightforward operation involving cosine functions.

Sine Function

The sine function, denoted as \(\sin\), is a fundamental trigonometric function that describes the relationship between an angle and the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. It is an odd function and is periodic with a period of \(2\pi\) radians (or 360 degrees), meaning it repeats its values in a regular cycle.

The sine function plays a pivotal role in various areas of mathematics, from defining the imaginary part of complex numbers to modeling periodic phenomena in physics, such as waves and oscillations. In the exercise above, we see the sine function being used with an angle variable \(\theta\), and the product of two such sine functions is effectively simplified using the product-to-sum formula.

Cosine Function

Similar to the sine function, the cosine function, denoted as \(\cos\), is another basic trigonometric function. It represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle when considering a particular angle. The cosine function is even and shares the same periodicity as the sine function, with a period of \(2\pi\) radians (360 degrees).

Cosine functions are used in a variety of mathematical contexts, such as the real part of complex numbers, the description of harmonic motions, and the calculation of distances in coordinate systems. In our exercise, after applying the product-to-sum formula, we obtained a sum involving cosine functions with different arguments, simplifying the initial product expression of sine functions and thus illustrating the cosine function's relevance in conjunction with the sine function.