Problem 99 - Algebra and Trigonometry

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

No, the equation \( f(x) = g(x) \) is not an identity; it holds only at specific points.

Step 1: Expand $f(x)$

Expand the expression for \( f(x) = (\sin x + \cos x)^2 \) using the formula \((a + b)^2 = a^2 + 2ab + b^2\). This gives:\[ f(x) = \sin^2 x + 2 \sin x \cos x + \cos^2 x\].

Step 2: Use the Pythagorean Identity

Recall that the Pythagorean Identity states \( \sin^2 x + \cos^2 x = 1 \). Substitute this identity into the expression for \( f(x) \). This gives:\[ f(x) = 1 + 2 \sin x \cos x\].

Step 3: Transform the Double Angle

Recognize that \( 2\sin x\cos x \) can be rewritten using the double angle identity for sine: \( 2\sin x \cos x = \sin(2x) \). Therefore, \( f(x) = 1 + \sin(2x) \).

Step 4: Compare $f(x)$ and $g(x)$

We have transformed \( f(x) \) into \( 1 + \sin(2x) \). Set it equal to \( g(x) \), which is simply \( 1 \):\[ 1 + \sin(2x) = 1\]. Simplifying this, results in \( \sin(2x) = 0 \).

Step 5: Analyze $f(x)=g(x)$ as an Identity

For \( f(x) \) to equal \( g(x) \) for all \( x \), the equation \( \sin(2x) = 0 \) must be true for all \( x \), which only holds for specific values. Therefore, \( f(x) = g(x) \) is not an identity for all values of \( x \).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions can give visual insights into relationships between them. In this exercise, we compare two functions, \( f(x) \) and \( g(x) \), by plotting their graphs on the same set of axes. This comparison helps us determine if the functions are equivalent for all input values, or if they overlap at certain points.

• Function \( f(x) \): Derived from \((\sin x + \cos x)^2\). It contains trigonometric components that vary based on the angle \( x \).
• Function \( g(x) \): A constant value of \( 1 \), represented as a straight line on the graph.

By graphing, we observe if the curves of \( f(x) \) are identical to that of \( g(x) \). Graphing visually suggests where \( f(x) = g(x) \); specifically, at points where they intersect. However, these intersections aren't continuous, indicating that the functions are equivalent only at specific \( x \) values, not universally.

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric principle. It states that for any angle \( x \), the sum of the squares of sine and cosine is equal to 1: \[ \sin^2 x + \cos^2 x = 1 \]. This identity is crucial for simplifying and transforming trigonometric expressions.
In our exercise, we use the Pythagorean Identity to simplify \( f(x) \):

• Original expression: \((\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x\).
• Apply the identity: Replace \( \sin^2 x + \cos^2 x \) with 1, simplifying to \( 1 + 2\sin x \cos x \).

This substitution streamlines the expression and makes the underlying patterns clearer, aiding in further analysis and transformations of trigonometric functions.

Double Angle Formulas

The Double Angle Formulas are a set of equations used to express trigonometric functions at twice an angle. They are essential tools in trigonometric transformations.
The specific formula relevant to our problem is for sine:

• \( 2 \sin x \cos x = \sin(2x) \).

In the given exercise, transforming \( 2 \sin x \cos x \) to \( \sin(2x) \) simplifies the expression for \( f(x) \) to \( 1 + \sin(2x) \).
Such transformations are potent because they convert complex trigonometric expressions into simpler forms, useable for further manipulation or comparison. In this scenario, this transformation directly assists in comparing \( f(x) \) with the constant function \( g(x) = 1 \).

Sine and Cosine Functions

Sine and cosine are core functions in trigonometry, describing the relationship of angles and sides in a right triangle. Their values oscillate between -1 and 1, creating a periodic wave-like graph.
A few key characteristics include:

• Sine Function: \( \sin(x) \) peaks when \( x = \frac{\pi}{2} + 2k\pi \) and troughs when \( x = \frac{3\pi}{2} + 2k\pi \), where \( k \) is an integer.
• Cosine Function: \( \cos(x) \) peaks at \( x = 2k\pi \) and troughs at \( x = \pi + 2k\pi \).

In practice, sine and cosine are frequently involved in expressions to form other trigonometric identities. Understanding their behavior is essential for predicting how they affect compound functions like \( f(x) = (\sin x + \cos x)^2 \).
Knowing that they are periodic helps anticipate their effects in expressions, leading to more effective graphing and identity verification.