Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the period and amplitude. $$y=5 \cos \frac{x}{2}$$

Step-by-Step Solution

Answer

The amplitude of the function is 5, and the period is \(4 \pi\).

Step 1: Identify the Amplitude

The amplitude of the function is the coefficient in front of the cosine function. In this case, the amplitude is 5.

Step 2: Identify the Coefficient of x in the argument

The coefficient of x in the argument of the cosine function affects the period. In this case, the coefficient of x is \( \frac{1}{2}\).

Step 3: Calculate the Period

The period of a cosine function in general form is given by \( \frac{2 \pi}{\text{coefficient of x}} \). The coefficient of x here is \( \frac{1}{2}\) , so the period is \( \frac{2 \pi}{\frac{1}{2}} = 4 \pi \).

Key Concepts

Amplitude

In trigonometric functions, the amplitude refers to the height of the wave. Specifically, it indicates how far away the waves of the function move from the center line, or equilibrium, before turning back.
For a cosine function, the amplitude is given by the absolute value of the coefficient of the cosine term.
For the function \(y = 5 \cos \frac{x}{2}\), the coefficient of the cosine is 5. This means the amplitude is \(|5| = 5\).

The amplitude affects how 'tall' or 'short' the wave appears on a graph.

• Higher amplitude means taller waves.
• Lower amplitude means shorter waves.

Understanding amplitude is essential as it affects the perceived intensity or strength of the wave in real-world applications, such as sound and light waves.

Period of Cosine Function

The period of a cosine function is the distance along the x-axis, which the function takes to complete one full cycle of its wave-like pattern.
For a basic cosine function, \(y = \cos(x)\), the period is \(2\pi\), representing one full oscillation from start to finish.
However, when the argument of the cosine is scaled by a coefficient (e.g., \(\frac{x}{2}\)), this affects the period.

The period of \(y = 5\cos \frac{x}{2}\) is determined by the formula:

• \[ \text{Period} = \frac{2\pi}{\text{coefficient of } x}\]
• With \(\frac{1}{2}\) as the coefficient in the argument, the period becomes \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).

As a result, this cosine function takes a longer interval along the x-axis to complete its cycle compared to the standard cosine function. Knowing the period is crucial in fields such as signal processing, where it helps understand frequencies.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often denoted by \(\cos(x)\).
It displays a repeating wave-like pattern as the angle \(x\) varies, which is characteristic of periodic functions.

The general form of a cosine function is:

• \( y = a \cos(bx + c) + d \)
• Where:
  • \(a\) is the amplitude.
  • \(b\) affects the period.
  • \(c\) results in a phase shift.
  • \(d\) results in a vertical shift.

For the function \(y = 5\cos \frac{x}{2}\), there are no phase or vertical shifts, keeping the wave centered on the x-axis, just amplified and stretched along the x-axis.
The cosine function is integral in mathematical applications involving oscillations, such as waves, circular motion, and more, allowing precise modeling and prediction of behaviors in physics and engineering.