Problem 70 - Contemporary Precalculus

Question

(a) Judging from their graphs, which of the functions \(f(t)=\sin t, g(t)=\cos t,\) and \(h(t)=\tan t\) appear to be even functions? Which appear to be odd functions? (b) Confirm your answers in part (a) algebraically by using appropriate identities from Section 6.3

Step-by-Step Solution

Answer

Answer: \(f(t)=\sin t\) and \(h(t)=\tan t\) are odd functions, whereas \(g(t)=\cos t\) is an even function.

Step 1: Identify even/odd functions using their graphs

First, let's check the given functions graphically to understand if they are even or odd functions. Graphs of these functions have the following properties; Even function: symmetrical with respect to the y-axis (i.e., \(f(-x) = f(x)\)) Odd function: symmetrical with respect to the origin (i.e., \(f(-x) = -f(x)\)) Looking at the graphs of the three functions: - \(\sin t\) is an odd function. - \(\cos t\) is an even function. - \(\tan t\) is an odd function. Now, let's confirm this algebraically.

Step 2: Confirm the results algebraically

To algebraically confirm the parity of these trigonometric functions, we will use the following identities from Section 6.3: \(\sin (-x) = -\sin x\) \(\cos (-x) = \cos x\) \(\tan (-x) = -\tan x\) These identities directly confirm our graphical analysis: - For \(f(t) = \sin t\), since \(\sin (-x) = -\sin x\), it's an odd function. - For \(g(t) = \cos t\), since \(\cos (-x) = \cos x\), it's an even function. - For \(h(t) = \tan t\), since \(\tan (-x) = -\tan x\), it's an odd function. In conclusion, based on the graphs and algebraic identities, \(f(t)=\sin t\) and \(h(t)=\tan t\) are odd functions, whereas \(g(t)=\cos t\) is an even function.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is akin to having a master key for the world of trigonometry. These identities are equations that hold true for any value of the variable within their domains. They allow us to simplify complex trigonometric expressions, solve trigonometric equations, and verify the parity (even or odd) of trigonometric functions.

For example, the following are fundamental identities for the sine, cosine, and tangent functions:

• \forall x, \(\sin (-x) = -\sin x\), reflecting that sine is an odd function.
• \forall x, \(\cos (-x) = \cos x\), indicating that cosine is an even function.
• \forall x, \(\tan (-x) = -\tan x\), showing that tangent is an odd function.

These identities crucially depict that for odd functions, flipping the input's sign 'flips' the output, while even functions maintain their output regardless of the input’s sign. This concept is not just algebraically neat—it resonates with the core idea of symmetry in functions.

Function Symmetry

Trigonometric functions, much like any mathematical function, can possess symmetries. There are two primary types: even and odd symmetries. These symmetries simplify understanding how functions behave and how they can be transformed or reflected.

An even function is mirrored across the y-axis. Mathematically, a function is even if \(f(-x) = f(x)\). In essence, if you take a point on the function and reflect it over the y-axis, it lands back on the function. Cosine is a classic example of this, as for any angle \(x\), \(\cos (-x) = \cos x\).

Identifying Odd Symmetry

Conversely, an odd function has rotational symmetry around the origin, meaning that rotating a point 180 degrees around the origin places it back onto the function. This is expressed as \(f(-x) = -f(x)\). The sine and tangent functions are illustrations of odd symmetry, as both follow the pattern that \(\sin (-x) = -\sin x\) and \(\tan (-x) = -\tan x\).These symmetries are not just theoretical constructs—they're visually and algebraically verifiable and essential for solving many practical problems in trigonometry.

Graphical Analysis of Functions

Graphical analysis allows us to visually understand a function's behavior, including its symmetry and periodicity, which can be more intuitive than algebraic methods. When analyzing the graphs of trigonometric functions, notice repetitive patterns that reveal their periodic nature—a fundamental characteristic of these functions.

When considering even and odd functions, their symmetry becomes apparent when graphed.

Even Function Graphs

For an even function such as \(g(t) = \cos t\), the graph is symmetrical along the vertical y-axis. This means if we were to 'fold' the graph along the y-axis, both sides would match perfectly, which is precisely what makes cosine an even function by definition.

Odd Function Graphs

For odd functions such as \(f(t) = \sin t\) and \(h(t) = \tan t\), the graph exhibits rotational symmetry about the origin. If we could rotate the right side of the graph 180 degrees around the origin, it would match the left side. Such visual cues are vital as they provide confirmation for the algebraic properties we use to define trigonometric functions.

Using graphical analysis, we can often predict the nature of a function before diving into algebraic proofs, as seen when first examining the trigonometric functions in the original exercise. However, it is the combination of both graphical and algebraic understanding that offers a robust approach to mastering trigonometric concepts and their applications in various mathematical contexts.