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Chapter 8: Problem 3

True or False The graph of \(y=\cos x\) is decreasing on the interval \([0, \pi]\)

Short Answer

Expert verified
True, the graph of \(y = \cos x\) is decreasing on the interval \([0, \pi]\).

Step by step solution

01

Understand the Cosine Function

The function given is the cosine function, represented by the equation \(y = \cos x\). The cosine function oscillates between -1 and 1 and has a fundamental period of \(2\pi\).
02

Identify the Interval

The interval to consider is \([0, \pi]\). This means examining the behavior of the function from \(x = 0\) to \(x = \pi\).
03

Analyze the Behavior of Cosine

The cosine function starts at \(y = 1\) when \(x = 0\) and decreases to \(y = -1\) when \(x = \pi\). This behavior shows a monotonic decrease on the interval \([0, \pi]\).
04

Conclude

Since the function \(y = \cos x\) consistently decreases from \(x = 0\) to \(x = \pi\), the statement is true.

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are vital in mathematics, particularly in understanding periodic phenomena. One such function is the cosine function, represented by the equation \( y = \cos x \). The cosine function maps an angle \( x \) to the horizontal coordinate of a point on the unit circle. It oscillates between -1 and 1 over a period of \( 2\pi \). This periodic nature makes the cosine function useful in diverse applications, including physics, engineering, and signal processing. Understanding this function's behavior is crucial for solving many problems in trigonometry.
Monotonic Decreasing
A function is said to be *monotonic decreasing* if it consistently decreases over a specific interval. For example, in the case of the cosine function \( y = \cos x \), within the interval \[ 0, \pi \], the function values decrease steadily. At \( x = 0 \), the cosine function starts at 1. As \text\{x\} increases towards \( \pi \), the value of \( \cos x \) decreases, reaching -1. This steady decrease without any increase makes the function monotonic decreasing in that interval.
Interval Analysis
Interval analysis involves examining a function's behavior within a particular range of its domain. For the cosine function \( y = \cos x \), we analyze the interval \[ 0, \pi \] to understand its behavior. In this interval, the function goes from 1 at \( x = 0 \) to -1 at \( x = \pi \). By considering smaller sub-intervals within \[ 0, \pi \], we can confirm that the function does not increase at any point, thereby confirming its monotonic decrease within the interval.
Periodicity
Periodicity refers to a function's repeating pattern over regular intervals. For trigonometric functions like cosine, this is a defining characteristic. The cosine function \( y = \cos x \) has a period of \( 2\pi \), which means it repeats every \( 2\pi \) units along the \( x \)-axis. This property helps us predict the function's behavior over extended intervals. Understanding periodicity is essential for identifying patterns and making predictions in various fields such as signal processing and harmonic analysis.

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Most popular questions from this chapter

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