Problem 4 - Contemporary Precalculus

Question

Find the rule of the product function fg. $$f(t)=\sin 2 t+\cos ^{4} t ; \quad g(t)=\cos 2 t+\cos ^{2} t$$

Step-by-Step Solution

Answer

Answer: The product function fg(t) is given by: fg(t) = sin(2t)⋅cos(2t) + sin(2t)⋅cos^2(t) + cos^4(t)⋅cos(2t) + cos^4(t)⋅cos^2(t).

Step 1: Identify the given functions

We are given the following functions: $$f(t)=\sin(2t) + \cos^{4}(t)$$ $$g(t)=\cos(2t) + \cos^{2}(t)$$

Step 2: Multiply f(t) and g(t)

To find the product function fg(t), we multiply both functions: $$fg(t) = f(t) \cdot g(t)$$

Step 3: Substitute the given functions and simplify the expressions

Now we will substitute the given functions into the expression and then simplify it: $$fg(t) = (\sin(2t) + \cos^{4}(t)) \cdot (\cos(2t) + \cos^{2}(t))$$

Step 4: Distribute the terms

We will now distribute the terms: $$fg(t) = \sin(2t) \cdot \cos(2t) + \sin(2t) \cdot \cos^{2}(t) + \cos^{4}(t) \cdot \cos(2t) + \cos^{4}(t) \cdot \cos^{2}(t)$$ Now we have found the rule of the product function fg(t): $$fg(t) = \sin(2t) \cdot \cos(2t) + \sin(2t) \cdot \cos^{2}(t) + \cos^{4}(t) \cdot \cos(2t) + \cos^{4}(t) \cdot \cos^{2}(t)$$

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential when dealing with functions that involve sine and cosine. These identities are like shortcuts that help simplify complex trigonometric expressions and solve problems more efficiently.

For instance, one of the most widely used trigonometric identities is the Pythagorean identity, which states that for any angle \( t \), \( \sin^2(t) + \cos^2(t) = 1 \). There are also double-angle formulas, like \( \sin(2t) = 2\sin(t)\cos(t) \) and \( \cos(2t) = \cos^2(t) - \sin^2(t)\), that are invaluable for simplifying expressions involving the product of trigonometric functions.

When you encounter a product of sinusoidal functions, such as \( \sin(2t)\cdot\cos(2t) \), knowing these identities can transform the expression into something more manageable or even into its fundamental components. This skill is key to making complex problems solvable and is a stepping stone to mastering trigonometry.

Function Multiplication

Function multiplication is a critical operation in mathematics that involves multiplying two functions together. It follows the distributive law, much like multiplying two binomials in algebra. In the context of the given problem, we are tasked with finding the product function \(fg(t)\) which is the result of multiplying \(f(t)\) and \(g(t)\).

When you multiply functions, you essentially take each term from the first function and multiply it by each term in the second function, a process known as distribution. For example, \(f(t)\) might be represented as \(a + b\) and \(g(t)\) as \(c + d\), then their product \(f(t)g(t)\) would be \(ac + ad + bc + bd\).

It's important to approach this operation systematically to avoid errors and to simplify the terms whenever possible. This applies whether the functions involve arithmetic operations, algebraic expressions, or, as in our case, trigonometric functions.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics, essential for making complex equations more manageable and understandable. The aim is to express the equation in its simplest form, while preserving its original value.

When simplifying the product of trigonometric functions, it is important to look for common factors, use trigonometric identities where possible, and reduce the expression step by step. This process may involve combining like terms, factoring, and canceling terms that appear in both the numerator and the denominator. By doing so, it can make the otherwise daunting trigonometric equations much more approachable.

In the context of our exercise, distributing the terms first and then applying trigonometric identities of sum and double-angle might reveal opportunities for simplification. By practicing these techniques, students become more adept at working with complex expressions and strengthen their problem-solving skills.