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Chapter 4: Problem 6

Finding the Period and Amplitude, find the period and amplitude. $$ y=3 \cos 2 x $$

Short Answer

Expert verified
The amplitude of the function \(y=3 \cos 2x\) is 3, and the period is \(\pi\).

Step by step solution

01

Identify the amplitude

The amplitude of the function is the absolute value of the coefficient before the cosine function. In our function \(y=3 \cos 2x\), the coefficient before the cosine function is 3, so the amplitude is|3| = 3.
02

Identify the period

The period of a cos function can be determined by evaluating \(2\pi\) divided by the absolute value of the coefficient of \(x\). In our function \(y=3 \cos 2x\), the coefficient of \(x\) is 2, so the period is \(2\pi/|2| = \pi\).

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Key Concepts

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

Understanding Amplitude in Trigonometric Functions
Amplitude is a core concept when dealing with trigonometric functions such as sine and cosine. It represents the height of the wave from its central axis to its peak. In simple terms, amplitude is the maximum distance the wave extends above or below its equilibrium.
To find the amplitude in a cosine function, look for the coefficient that multiplies the cosine term. For the function \(y = 3 \cos 2x\), this coefficient is 3. Hence, the amplitude is \(|3| = 3\).
  • The amplitude is always a non-negative value.
  • It affects how "tall" or "short" the function appears on a graph.
  • A larger amplitude means taller waves, while a smaller one indicates shorter waves.
Exploring the Period of a Cosine Function
The period of a trigonometric function is the interval over which it repeats itself. For cosine functions, the period is determined by the formula \(\frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(x\) in the function \(y = a \cos(bx)\).
In the given function \(y = 3 \cos 2x\), the coefficient of \(x\) is 2. Thus, the period of this function is calculated as \(\frac{2\pi}{2} = \pi\). This means that the wave repeats every \(\pi\) units along the x-axis.
  • A smaller period implies the wave cycles more frequently over a given interval.
  • The "tightness" of the wave corresponds to how many cycles are packed within a particular horizontal distance.
  • Interestingly, changing the period does not affect the amplitude.
The Nature of the Cosine Function
The cosine function is a fundamental trigonometric function used to model periodic phenomena. It is defined over the domain of all real numbers and has a unique characteristic that defines it: its wave-like pattern.
In its basic form, \(y = \cos x\), the wave oscillates between -1 and 1, with a period of \(2\pi\). The graph of the cosine function starts at its maximum value, decreases to its minimum, and returns to its maximum over one complete cycle.
When transformed into \(y = a \cos(bx)\):
  • The amplitude \(|a|\) affects its vertical stretch.
  • The period \(\frac{2\pi}{|b|}\) compresses or elongates it horizontally.
  • This function is particularly useful in physics and engineering to describe oscillations, sound waves, and light waves.
Altering these parameters allows for the simulation of various real-world cyclical behaviors using the cosine function.

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Most popular questions from this chapter

Find the exact value of the expression. (Hint: Sketch a right triangle.) \(\sec \left[\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\right]\)

Use a graphing utility to graph the function. \(f(x)=2 \arccos (2 x)\)

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x .\) \(g(x)=\arcsin (x-1)\)

Use a graphing utility to graph the function. \(f(x)=\arctan (2 x-3)\)

Use a calculator to evaluate the expression. Round your result to two decimal places. \(\arcsin \frac{7}{8}\)
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