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Chapter 4: Problem 94

Without drawing a graph, describe the behavior of the basic cosine curve.

Short Answer

Expert verified
The basic cosine curve is a wave that oscillates between -1 and 1, exhibits a period of \(2\pi\) (around 360掳) and a frequency of \(1/(2\pi)\), or approximately 0.159 per degree. Key points include maximums at \(0\) and \(2\pi\), zeros at \(\pi/2\) and \(3\pi/2\), and a minimum at \(\pi\).

Step by step solution

01

Define Basic Shape

Cosine is a periodic function that has a wave-like shape and repeats every \(2\pi\) (around 360掳). The basic shape starts at a maximum, goes down to a minimum, goes back up to the same maximum, and then repeats.
02

Discuss Amplitude

The amplitude of the cosine function is 1. This is the maximum absolute value it reaches, both minimum and maximum. In other words, the function oscillates between -1 and 1.
03

Identify Period and Frequency

The period of the basic cosine function is \(2\pi\) (around 360掳), meaning it completes one full wave within this range. The frequency is the reciprocal of the period, which is \(1/(2\pi)\) or approximately 0.159 per degree.
04

Highlight Important Points

The cosine function has its maximum value 1 at \(0\) and \(2\pi\); it is equal to \(0\) at \(\pi/2\) and \(3\pi/2\); and has its minimum value -1 at \(\pi\).

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