Problem 56 - Calculus of a Single Variable

Question

Comparing Functions In Exercises 55 and \(56,\) use symmetry, extrema, and zeros to sketch the graph of \(f .\) How do the functions \(f\) and \(g\) differ? $$ \begin{array}{l}{f(t)=\cos ^{2} t-\sin ^{2} t} \\ {g(t)=1-2 \sin ^{2} t}\end{array} $$

Step-by-Step Solution

Answer

The functions \(f(t)\) and \(g(t)\) do not differ, they are the same. Both functions have been simplified and rewritten as \(\cos(2t)\), showing that they are identical in terms of their period, amplitude, frequency, phase shift, vertical shift, and symmetry.

Step 1: Analyzing the function \(f(t) = \cos^2 t - \sin^2 t\)

First, simplify the function using trigonometric identities. The function \(f(t) = \cos^2 t - \sin^2 t\) takes advantage of the fundamental trigonometric identity \(cos(2t) = cos^2 t - sin^2 t\). So, the function \(f(t)\) can be written as \(f(t) = \cos(2t)\). The period of this function is determined by \(2\pi/|2|\), which is \(\pi\). The function has symmetry around the y-axis, meaning it is an even function.

Step 2: Analyzing the function \(g(t) = 1 - 2\sin^2 t\)

The function can be rewritten using the double-angle identity for cosine, \(cos(2t) = 1 - 2sin^2 t\). So, the function \(g(t) = \cos(2t)\) is equivalent to \(f(t)\). The period of this function is \(\pi\), and the function is an even function, showing symmetry around the y-axis.

Step 3: Comparing functions \(f(t)\) and \(g(t)\)

From the analysis in Steps 1 and 2, we can see that \(f(t) and \(g(t)\) are identical functions. Both functions are even, have the same period of \(\pi\), and they have the same graph. Therefore, there is no difference between \(f(t)\) and \(g(t)\).

Key Concepts

Symmetry in Functions

Understanding symmetry in functions is essential for analyzing trigonometric functions like \(f(t) = \cos^2 t - \sin^2 t\). Symmetry can reveal important characteristics of a function, such as being even or odd. An even function, like the ones in this exercise, satisfies \(f(-t) = f(t)\), naturally exhibiting symmetry about the y-axis. This means the left half of the graph mirrors the right half, making it easier to sketch.
Odd functions, on the other hand, satisfy \(f(-t) = -f(t)\) and are symmetric about the origin. Identifying symmetry helps reduce the complexity when working with graphing and integration. Recognize these patterns in trigonometric functions to make sense of their graphs quickly.

Trigonometric Identities

Trigonometric identities are powerful tools that simplify complex trigonometric expressions. In this exercise, we utilized the double-angle identity. It states:
\[ \cos(2t) = \cos^2 t - \sin^2 t \]
Using this, we transformed \(f(t) = \cos^2 t - \sin^2 t\) into \(f(t) = \cos(2t)\).

Another related identity used is:
\[ 1 - 2\sin^2 t = \cos(2t) \]
It shows how \(g(t) = 1 - 2\sin^2 t\) is equivalent to \(f(t) = \cos(2t)\).
Exploring these identities reveals deep interconnections between the trigonometric functions. They allow us to simplify, compare, and analyze functions effectively. Always keep these identities handy to tackle trigonometric problems with ease.

Periodicity of Functions

Periodicity is a fundamental aspect of trigonometric functions, describing how they repeat at regular intervals. In this exercise, both functions, transformed to \(\cos(2t)\), share a period of \(\pi\).

The period of a trigonometric function like \(\cos(kt)\) is given by \(\frac{2\pi}{|k|}\). Here, since \(k = 2\), the period becomes \(\pi\).

• Understanding periodicity helps you predict the behavior of functions over different intervals.
• It's crucial for both sketching graphs and solving equations.
• Knowing the period allows us to verify that functions repeat their values predictably and regularly.

The periodic nature of trigonometric functions makes them incredibly useful in modeling cyclical phenomena part of real-world applications.