Problem 97 - 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. $$f(x)=\cos ^{2} x-\sin ^{2} x, \quad g(x)=1-2 \sin ^{2} x$$

Step-by-Step Solution

Answer

Yes, the graphs suggest an identity; proving algebraically shows \(f(x) = g(x)\) for all \(x\).

Step 1: Graph the Functions

To determine if \(f(x)\) and \(g(x)\) are identical, we should start by graphing the functions. \[ f(x) = \cos^2 x - \sin^2 x \]\[ g(x) = 1 - 2\sin^2 x \]Graph these functions in a domain, such as \([-2\pi, 2\pi]\). Use graphing software or a graphing calculator to plot them as they have trigonometric components.

Step 2: Analyze Graphs

Upon graphing, observe whether the graphs of \(f(x)\) and \(g(x)\) appear to overlap completely over the chosen interval. If they do, this suggests that the equation \(f(x) = g(x)\) might be an identity.

Step 3: Verify Algebraically

To confirm if \(f(x) = g(x)\) is an identity, equate them and simplify:\[ \cos^2 x - \sin^2 x = 1 - 2\sin^2 x \]Utilize the Pythagorean identity \( \cos^2 x + \sin^2 x = 1 \) to rewrite one of the trigonometric identities if needed.Substitute \( \cos^2 x = 1 - \sin^2 x \) into \(f(x)\):\[ (1 - \sin^2 x) - \sin^2 x = 1 - 2\sin^2 x \]This simplifies to:\[ 1 - 2\sin^2 x = 1 - 2\sin^2 x \]The equation holds for all \(x\), proving that \(f(x) = g(x)\) is indeed an identity.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions like sine, cosine, and their squared counterparts can be easily done using a graphing calculator or software. These tools allow us to visualize the behavior of the functions over specific intervals. For our specific example, we graphed the functions

• \( f(x) = \cos^2 x - \sin^2 x \)
• \( g(x) = 1 - 2\sin^2 x \)

When you graph these functions in the interval \([-2\pi, 2\pi]\), you'll observe how they behave over multiple periods.

It's important to choose a viewing window that effectively showcases the periodic nature of these trigonometric functions. Often, using an interval that encompasses several periods (like \([-2\pi, 2\pi]\)) can show their repeating patterns more clearly. By seeing the graphs overlap completely, it initially suggests that the two functions may indeed represent the same expression, leading us to explore this possibility further through algebraic verification.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry, expressing the relationship between the sine and cosine functions. It states:\[\cos^2 x + \sin^2 x = 1\]This identity is pivotal when simplifying trigonometric expressions, particularly when one needs to express cosine or sine squared in terms of the other.

In the exercise given, we used this identity to aid in proving the equation \( f(x) = g(x) \). Specifically, we expressed \( \cos^2 x \) as \( 1 - \sin^2 x \) and substituted it into the expression for \( f(x) \). By doing this, we showed that \[(1 - \sin^2 x) - \sin^2 x = 1 - 2\sin^2 x\]simplifies to \[1 - 2\sin^2 x = 1 - 2\sin^2 x\]This verification confirmed that \( f(x) \) and \( g(x) \) are indeed identical, reinforcing the usefulness of the Pythagorean identity in proving equivalence in trigonometric expressions.

Cosine and Sine Functions

The cosine and sine functions are essential in trigonometry and represent orthogonal projections of a unit circle on the xy-plane. They are periodic functions with a period of \(2\pi\), which means they repeat their values in regular intervals.

Both functions are fundamental in understanding wave-like behaviors as they arise repeatedly in applications involving periodic phenomena.

• The cosine function, \( \cos x \), starts at 1 when \( x = 0 \) and decreases, reaching its minimum at \( \pi \).
• The sine function, \( \sin x \), starts at 0 and reaches its maximum at \( \pi/2 \).

In the context of our exercise, knowing how squares of cosine and sine relate through the Pythagorean identity allows simplification of more complex trigonometric equations. By subvocalizing these relationships, we can manipulate and verify identities, like transforming \( \cos^2 x \) using the equation \( \cos^2 x = 1 - \sin^2 x \). Such transformations help investigate and confirm if two trigonometric functions are identical, as we demonstrated in proving \( f(x) = g(x) \).