Problem 78 - Precalculus

Question

For the following exercises, use a graph to determine whether are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think \(2 x=x+x . )\) $$ f(x)=\sin (4 x), g(x)=\sin (5 x) \cos x-\cos (5 x) \sin x $$

Step-by-Step Solution

Answer

The functions \( f(x) \) and \( g(x) \) are the same: \( \sin(4x) \).

Step 1: Identify the Functions

We have two functions given: \( f(x) = \sin(4x) \) and \( g(x) = \sin(5x)\cos(x) - \cos(5x)\sin(x) \). We need to determine if these functions are the same or different by simplifying or re-expressing \( g(x) \).

Step 2: Use Trigonometric Identity

Recognize that \( g(x) \) can be expressed using the trigonometric identity for sine: \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \). With \( a = 5x \) and \( b = x \), \( g(x) = \sin(5x - x) = \sin(4x) \).

Step 3: Compare Both Functions

Having simplified \( g(x) \), it becomes \( \sin(4x) \), which is identical to \( f(x) = \sin(4x) \). Thus, both functions, \( f(x) \) and the simplified \( g(x) \), are the same.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in solving and simplifying trigonometric expressions. They are equations that hold true for all values of the included variables. In trigonometry, one commonly used identity is the sine difference identity: \[ \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \] This identity allows us to simplify expressions by transforming complex trigonometric expressions into simpler ones. For instance, in the exercise provided, we had to simplify the function \( g(x) = \sin(5x)\cos(x) - \cos(5x)\sin(x) \). By applying the sine difference identity:

• Identify \( a = 5x \) and \( b = x \).
• Transform \( g(x) \) into \( \sin(5x - x) \).
• This further simplifies to \( \sin(4x) \).

Thus, trigonometric identities like this help simplify the analysis and comparison of trigonometric functions by providing alternate and sometimes more accessible forms of functions.

Function Comparison

Function comparison involves analyzing mathematical functions to determine if they are equivalent or not. In our exercise, the task was to determine if \( f(x) = \sin(4x) \) is the same as \( g(x) \). To do this:

• Simplify \( g(x) \) using trigonometric identities, like we did with the sine difference identity.
• Check the final simplified form of \( g(x) \) to see if it matches \( f(x) \).

Once we simplified \( g(x) \) to \( \sin(4x) \), it became apparent that \( f(x) \) and \( g(x) \) are equivalent. This comparison often involves employing identities or graphing both functions to visually determine if their graphs overlap or not. Function comparison can help validate mathematical statements by confirming the equality of two expressions.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that model periodic phenomena, such as sound waves or the motion of pendulums. They can be represented as \( \sin(x) \) and \( \cos(x) \). These functions have specific properties:

• Periodicity: Both sine and cosine functions repeat their values in cycles, typically every \( 2\pi \) radians.
• Amplitude: The amplitude of both functions is 1, meaning their values range between -1 and 1.

In the problem discussed, \( \sin(4x) \) is a modification of the basic sine function, with its period reduced by a factor of 4, resulting in more frequent oscillations. Understanding how sine and cosine functions behave, both individually and when they interact, such as in the expression \( g(x) \), allows us to make precise transformations and comparisons, as seen when we applied the sine difference identity to determine that \( g(x) \) was equivalent to a simple sine function.