Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the period and amplitude. $$y=\frac{1}{4} \cos \frac{4 x}{3}$$

Step-by-Step Solution

Answer

The amplitude is \(\frac{1}{4}\) and the period is \(\frac{3\pi}{2}\).

Step 1: Identify the Amplitude and Period

In the given function \(y=\frac{1}{4} \cos \frac{4x}{3}\), the amplitude is \(A = \frac{1}{4}\) and the coefficient of \(x\) inside the cosine function is \(B = \frac{4}{3}\).

Step 2: Compute the Amplitude

The amplitude is the absolute value of \(A\), hence it is \(\frac{1}{4}\).

Step 3: Compute the Period

The period can be calculated using the formula \(T = \frac{2\pi}{B}\). Here \(B = \frac{4}{3}\), so the period \(T = \frac{2\pi}{\frac{4}{3}} = \frac{3\pi}{2}.\)

Key Concepts

Amplitude

The amplitude of a trigonometric function tells us how much the waves of the function stretch or compress vertically. It's like measuring how tall the waves are from their average position. Amplitude is always a positive number, even if the function's coefficient is negative. In simple terms, it tells us the half-distance between the highest and lowest points of the wave.

For the function you're working with, the amplitude is determined by looking at the coefficient in front of the cosine function. If the function is written as \( y = A \cos(Bx) \), then the amplitude is \( |A| \). In this specific case, where your function is \( y = \frac{1}{4} \cos \left( \frac{4x}{3} \right) \), the amplitude is the absolute value of \( \frac{1}{4} \), which is \( \frac{1}{4} \).

This means the graph oscillates 0.25 units above and below the horizontal axis, reflecting smaller ripples in the wave, as the coefficient is less than 1.

• This small amplitude results in a less pronounced deviation from the center line.
• Even with variations to the period, the amplitude remains unaffected by changes in \( B \).

Period of a Function

The period of a trigonometric function tells us the length of one full cycle of the wave before it begins to repeat. It's like drawing one complete loop of a roller coaster before it starts all over again. This period is crucial for understanding the rhythm and frequency of the wave.

Using the formula for the period of any function in the form of \( y = A \cos(Bx) \), the period can be calculated as \( T = \frac{2\pi}{B} \). This formula essentially measures how "stretched out" the wave is along the x-axis.

In your exercise, where \( B = \frac{4}{3} \), we substitute into the formula:
\[ T = \frac{2\pi}{\frac{4}{3}} = \frac{3\pi}{2} \]
This result indicates that every \( \frac{3\pi}{2} \) units along the x-axis, the wave will repeat its pattern.

• A function with a longer period will complete fewer cycles within a given space.
• Shifts in the period modify the horizontal stretch, making it "wider" or "narrower" on the graph.

Cosine Function

The cosine function, similar to the sine function, is a fundamental periodic wave often used in trigonometry. It describes a smooth oscillating motion and is defined as \( y = \cos(x) \) in its simplest form.

The key characteristics of the cosine function include:

• Symmetry: Cosine is an even function, meaning \( \cos(-x) = \cos(x) \), which ensures symmetry about the y-axis.
• Range: It always outputs values between -1 and 1, considering a standard cosine function \( y = \cos(x) \).

In the exercise, your function is \( y = \frac{1}{4} \cos \left( \frac{4x}{3} \right) \). Here:

• The amplitude modifies the output range to \(-\frac{1}{4}\) to \(\frac{1}{4}\).
• The wave completes one cycle every \( \frac{3\pi}{2} \) units, as determined by the period \( \frac{3\pi}{2} \).

The role of cosine in trigonometric identities and transformations makes it a versatile tool in mathematics, evident by its part in waveform modeling like x-ray transmission or sound waves.