Problem 99 - Precalculus Mathematics for Calculus

Question

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=(\sin x+\cos x)^{2}, \quad g(x)=1$$

Step-by-Step Solution

Answer

The equation \( f(x) = g(x) \) is not an identity since \( f(x) = 1 + \sin(2x) \); \( f(x) \) is not equal to 1 for all \( x \).

Step 1: Understand the Functions

First, rewrite the function \( f(x) \) from given: \( f(x) = (\sin x + \cos x)^2 \). The function \( g(x) \) is simply \( g(x) = 1 \). Before graphing, simplify \( f(x) \) to compare more easily to \( g(x) \).

Step 2: Simplify the Function \( f(x) \)

Simplify \( f(x) \) by expanding the square: \[ f(x) = (\sin x + \cos x)^2 = \sin^2 x + 2 \sin x \cos x + \cos^2 x \]Using the Pythagorean identity, \( \sin^2 x + \cos^2 x = 1 \), the function becomes: \[ f(x) = 1 + 2 \sin x \cos x \]Further simplify \( 2 \sin x \cos x \) using the double angle identity: \[ 2 \sin x \cos x = \sin(2x) \]So, \( f(x) = 1 + \sin(2x) \).

Step 3: Compare Simplified \( f(x) \) and \( g(x) \)

Now that we have \( f(x) = 1 + \sin(2x) \) and \( g(x) = 1 \), compare the two functions. Notice that \( \sin(2x) \) is not always zero, which suggests that \( f(x) eq g(x) \) for all \( x \).

Step 4: Graph the Functions

Graph the functions \( f(x) = 1 + \sin(2x) \) and \( g(x) = 1 \) in the same viewing rectangle. You will observe that \( f(x) \) oscillates above and below \( g(x) \), confirming visually that they are not identical.

Step 5: Conclude with the Graphing Suggestion

Based on your graph, determine if \( f(x) = g(x) \) for all \( x \). Since \( f(x) \) oscillates and \( g(x) \) is a constant, the graph shows that they are not equal for all values, hence the equation is not an identity.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is key when dealing with such problems. Trigonometric identities are equations that hold true for all the angles involved. They are essential tools for simplifying expressions and solving equations.

The most commonly used identity in this exercise is the Pythagorean identity:

• \[ \sin^2 x + \cos^2 x = 1 \]

This identity reflects the fundamental relationship between the sine and cosine of any angle. By expressing the square of sine and cosine in this manner, we can often simplify trigonometric expressions, as seen in the steps provided for the function \( f(x) \).

Besides the Pythagorean identity, recognizing other identities like the double angle identity helps in transforming complicated expressions to more manageable forms. This process plays an integral role in comparing and understanding the behavior of trigonometric functions.

Function Simplification

Function simplification is a crucial concept in making problems more approachable and easier to solve. Simplifying expressions reduces their complexity, revealing the underlying relationships between different variables.

Consider the function \( f(x) = (\sin x + \cos x)^2 \). At first glance, this looks complicated. However, by expanding the square, we break it down into simpler terms:

• First, expand \( (\sin x + \cos x)^2 \) to get \( \sin^2 x + 2 \sin x \cos x + \cos^2 x \).
• Next, employ the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \), resulting in \( f(x) = 1 + 2 \sin x \cos x \).
• Finally, simplify further by applying the double angle identity.
  
  

Such simplifications not only make calculations more manageable but also enable more straightforward comparisons between functions, as achieved here between \( f(x) \) and \( g(x) \), proving they are not identical. Proper simplification is essential for obtaining accurate results and understanding the true nature of functions.

Double Angle Formula

The double angle formula is another important tool used to simplify trigonometric expressions. This formula is used to express trigonometric functions of double angles related to the original function. It plays a significant role in breaking down complex expressions involving multiple trigonometric terms into simpler components.

In this context, the double angle formula for sine, \( \sin(2x) = 2 \sin x \cos x \), allowed for the simplification of the double product term \( 2 \sin x \cos x \) in our expression of \( f(x) \). This conversion made it possible to rewrite \( f(x) \) as \( 1 + \sin(2x) \), which is much simpler to work with.

By utilizing the double angle formula:

• We convert problematic expressions into easier, well-known trigonometric functions, such as sine and cosine of double angles.
• We streamline the comparison between trigonometric expressions and functions.

Understanding and applying these formulas help in both visualizing and analyzing relationships between trigonometric functions.