Problem 79 - 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 \(2 x=x+x . )\) $$ f(x)=\sin (2 x), g(x)=2 \sin x \cos x $$

Step-by-Step Solution

Answer

The functions are the same due to the identity \(\sin(2x) = 2\sin(x)\cos(x)\).

Step 1: Understanding the Functions

We have two functions given: \(f(x) = \sin(2x)\) and \(g(x) = 2\sin(x)\cos(x)\). Our task is to determine whether these two functions are the same by using their graphs.

Step 2: Using Trigonometric Identity

Let's first use the trigonometric identity for the double angle formula for sine: \(\sin(2x) = 2\sin(x)\cos(x)\). This identity is important because it shows that \(f(x) = \sin(2x)\) can be rewritten as \(f(x) = 2\sin(x)\cos(x)\).

Step 3: Comparing the Functions

Now, compare \(f(x) = 2\sin(x)\cos(x)\) and \(g(x) = 2\sin(x)\cos(x)\). Since both expressions are identically equal through the identity used, it shows that \(f(x)\) and \(g(x)\) are the same.

Step 4: Conclusion from Graphs and Identity

Both functions, \(f(x)\) and \(g(x)\), when graphed, represent the same curve because they are equivalent expressions mathematically. The trigonometric identity verifies that they will always yield the same output for any input \(x\).

Key Concepts

Double Angle Formulas

The double angle formulas are a fascinating part of trigonometry that help simplify expressions involving trigonometric functions. One of the most commonly used double angle formulas is for sine: \( \sin(2x) = 2\sin(x)\cos(x) \). This formula is especially handy when you need to rewrite or simplify expressions in trigonometric identities or when solving equations.
When you have a problem that involves \( \sin(2x) \), it's beneficial to recognize that you can express it using \( \sin(x) \) and \( \cos(x) \). This change often makes calculations easier and reveals function equivalences that are not immediately obvious.

• These formulas can help transform a complex trigonometric expression into a more manageable form.
• They are especially useful in calculus when finding derivatives or integrals of trigonometric functions.
• When graphing, these identities reveal insights into how the function behaves over intervals.

Applying the double angle formula helped us realize that \( f(x) = \sin(2x) \) is the same as \( g(x) = 2\sin(x)\cos(x) \), showing how powerful and revealing these identities can be.

Function Equivalence

Determining whether two functions are equivalent involves showing that they yield the same outputs for all inputs in their domain. In our exercise, we examined whether \( f(x) = \sin(2x) \) and \( g(x) = 2\sin(x)\cos(x) \) are equivalent. Using trigonometric identities is a common method for verifying function equivalence.
The primary idea is to transform one of the functions using known identities or formulas to match the other function. For example, by applying the double angle identity for sine, you can rewrite \( \sin(2x) \) as \( 2\sin(x)\cos(x) \).

• This shows that both expressions calculate the same result for any value of \( x \), showing they are indeed equivalent.
• Understanding function equivalence is crucial in mathematics, as it helps identify alternate expressions and simplifications.
• It also assists in solving equations, optimizing expressions, and even validating results in applied problems.

In many contexts, recognizing function equivalence can simplify problems significantly, as it allows us to choose the most convenient form of a function for our calculations.

Graphical Analysis

Graphical analysis involves studying the graphical representation of functions to gain insights into their properties and behavior. It's a powerful technique to verify function equivalence visually, as identical functions will produce the same graph.
For our functions \( f(x) = \sin(2x) \) and \( g(x) = 2\sin(x)\cos(x) \), plotting the graphs of these functions can immediately show their equivalence. Both functions will trace the same waveform, confirming they are identical.

• Graphical analysis can reveal symmetries, periodicity, and other characteristics that mathematical expressions alone might not immediately show.
• It helps to visualize potential problems or confirm conclusions derived algebraically.
• The visual nature of graphing is also helpful for many learners, providing an intuitive understanding of relationships between functions.

In addition to confirming function equivalence, graphs are essential for predicting behaviors and values, making them an indispensable tool in many fields of mathematics and science.