Problem 1 - Contemporary Precalculus

Question

Find the rule of the product function fg. $$f(t)=3 \sin t ; \quad g(t)=\sin t+2 \cos t$$

Step-by-Step Solution

Answer

Answer: The rule of the product function (fg)(t) is 3sin^2(t) + 6sin(t)cos(t).

Step 1: Write the product function

Write down the product function as \((fg)(t) = f(t) \cdot g(t)\), where \(f(t)\) and \(g(t)\) are the functiof(t) = 3 \sin t ns given in the exercise: $$(fg)(t) = (3 \sin t)(\sin t + 2 \cos t)$$

Step 2: Distribute the multiplication

Distribute the multiplication of \(3 \sin t\) to both terms of the function \(g(t)\): $$(fg)(t) = (3 \sin t)(\sin t) + (3 \sin t)(2 \cos t)$$

Step 3: Simplify the product function

Simplify the product function by multiplying the terms: $$(fg)(t) = 3 \sin^2 t + 6 \sin t \cos t$$ This is the final rule of the product function \((fg)(t)\).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are very common in mathematics, especially in fields like engineering, physics, and astronomy. The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each function has unique characteristics and applications.

• Sine (sin): This function gives the ratio of the length of the side of the triangle opposite the angle to the hypotenuse.
• Cosine (cos): This function provides the ratio of the adjacent side's length to the hypotenuse.
• Tangent (tan): Tangent is the ratio of the sine to the cosine functions, or opposite to adjacent side.

Understanding these functions is crucial when dealing with waveforms, circular motion, and oscillations. They form the basis for solving real-world problems involving periodic phenomena.

Exploring Sin and Cos Functions

The sine and cosine functions are among the most important trigonometric functions. They are periodic functions and are fundamental in the study of oscillatory systems such as sound and light waves.

• Sin Function: Represented as \( \sin t \), this function oscillates between -1 and 1. It's a continuous function often used to model periodic motion.
• Cos Function: Represented as \( \cos t \), this function similarly oscillates between -1 and 1 but is phase-shifted by 90 degrees compared to the sine function. It also models periodicity.

These functions are critical in understanding the relationships between different points in a cycle. They are incredibly useful in graphing and predicting the behavior of periodic functions. In the original problem, both sine and cosine are used in multiplying to form a new composite function.

Understanding Function Multiplication

Function multiplication involves taking two or more functions and creating a single function by multiplying their outputs wherever defined. This is often referred to as the product of functions. The resulting function represents the combination of effects from both original functions.

• Start with two functions, \( f(t) \) and \( g(t) \).
• Multiply them to create a new function: \((fg)(t) = f(t) \cdot g(t)\).
• This new function, \( (fg)(t) \), will capture the interaction between \( f(t) \) and \( g(t) \) at any given \( t \).

Function multiplication is a powerful tool in calculus and algebra that allows us to simplify complex expressions and find new relationships. In the provided exercise, we see how multiplying the sine and cosine terms results in an expression that reveals deeper relationships through simplification.