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Chapter 7: Problem 80

Determine whether the function is even, odd, or neither. \(f(x)=\cos (\sin x)\)

Short Answer

Expert verified
The function \( f(x) = \cos(\sin x) \) is even.

Step by step solution

01

Define function types

First, let's remember the definitions. A function is even if \( f(-x) = f(x) \) for all \( x \) in its domain and odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition is met, the function is neither even nor odd.
02

Apply even function test

Now, test the function \( f(x) = \cos(\sin x) \). Calculate \( f(-x) \): \( f(-x) = \cos(\sin(-x)) \). Since \( \sin(-x) = -\sin(x) \), it follows that \( f(-x) = \cos(-\sin x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Parity
Function parity is a fundamental property that helps classify functions based on their symmetry. Understanding this can simplify many mathematical processes. There are two main types of function parity: even and odd functions.

  • **Even Functions**: A function is considered even if it satisfies the condition that for every input \( x \) in its domain, \( f(-x) = f(x) \). This implies that the graph of an even function is symmetric with respect to the y-axis.
  • **Odd Functions**: Conversely, a function is classified as odd if \( f(-x) = -f(x) \) holds true for all \( x \) in its domain. The graph of an odd function is symmetric with respect to the origin.
  • **Neither**: If neither condition is met, the function is neither even nor odd, which means it lacks this particular symmetry.

These definitions are integral in various fields, particularly in Fourier analysis, where even and odd function properties simplify calculations.
Trigonometric Functions
Trigonometric functions are functions related to angles and are typically periodic. The basic trigonometric functions include sine, cosine, and tangent among others. Recognizing the parity of trigonometric functions can aid in understanding more complex expressions.

  • **Sine Function \( \sin(x) \)**: This function is odd, which means \( \sin(-x) = -\sin(x) \). Its graph exhibits origin symmetry, making it useful for modeling periodic phenomena.
  • **Cosine Function \( \cos(x) \)**: Cosine is an even function since \( \cos(-x) = \cos(x) \). It shows symmetry along the y-axis and is also pivotal in wave-based applications.

Trigonometric functions play a crucial role in geometry, physics, and engineering.
Cosine Function
The cosine function, denoted by \( \cos(x) \), is one of the most fundamental trigonometric functions. It is periodic with a period of \( 2\pi \), meaning its pattern repeats every \( 2\pi \) units.

  • **Even Nature**: As an even function, \( \cos(x) \) satisfies \( \cos(-x) = \cos(x) \). Its y-axis symmetry simplifies many calculations in trigonometry and calculus.

When considering the function \( \cos(\sin x) \), we involve a composition of functions. Since \( \sin(x) \) is odd, \( \sin(-x) = -\sin(x) \), causing a shift in symmetry when input into \( \cos(x) \). Consequently, this impacts the parity of \( \cos(\sin x) \), potentially making it neither purely even nor odd. Understanding these relationships aids in deeper analysis and application of trigonometric functions.

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Most popular questions from this chapter

Find a function that models the simple harmonic motion having the given properties. Assume that the displacement is at its maximum at time \(t=0\) . amplitude \(35 \mathrm{cm}, \quad\) period 8 \(\mathrm{s}\)

Spring-Mass System The frequency of oscillation of an object suspended on a spring depends on the stiffness \(k\) of the spring (called the spring constant) and the mass \(m\) of the object. If the spring is compressed a distance \(a\) and then allowed to oscillate, its displacement is given by $$ f(t)=a \cos \sqrt{k / m} t $$ (a) A 10 -g mass is suspended from a spring with stiffness \(k=3 .\) If the spring is compressed a distance 5 \(\mathrm{cm}\) and then released, find the equation that describes the oscillation of the spring. (b) Find a general formula for the frequency (in terms of \(k\) and \(m ) .\) (c) How is the frequency affected if the mass is increased? Is the oscillation faster or slower? (d) How is the frequency affected if a stiffer spring is used (larger \(k\) )? Is the oscillation faster or slower?

Clock Pendulum The pendulum in a grandfather clock makes one complete swing every 2 s. The maximum angle that the pendulum makes with respect to its rest position is \(10^{\circ} .\) We know from physical principles that the angle \(\theta\) between the pendulum and its rest position changes in simple harmonic fashion. Find an equation that describes the size of the angle \(\theta\) as a function of time. (Take \(t=0\) to be a time when the pendulum is vertical)

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$ y=1.6 \sin (t-1.8) $$

The given function models the displacement of an object moving in simple harmonic motion. (a) Find the amplitude, period, and frequency of the motion. (b) Sketch a graph of the displacement of the object over one complete period. $$ y=2.4 \sin 3.6 t $$
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