Problem 6 - Precalculus: A Concise Course

Question

Find the period and amplitude. $$ y=3 \cos 2 x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Amplitude A

The amplitude, denoted by A is the coefficient of the cosine function. From the given equation \(y = 3 \cos 2x\), it's seen that A = 3 is the amplitude of the function.

Step 2: Identify the coefficient B and calculate Period

The coefficient in front of x inside the cosine function is B, which affects the period of the function. The period of a cosine function is given by \(2\pi/B\). From the given function \(y = 3 \cos 2x\), we see that B = 2. Therefore, the period is \(2\pi/2 = \pi\) .

Key Concepts

Amplitude

When we talk about the amplitude of a trigonometric function, like the cosine function, we're referring to how far the peaks and valleys of the graph reach from the central axis. The amplitude is a measure of the wave's height. In the context of the function \(y = 3\cos 2x\), the amplitude is represented by the coefficient that appears in front of the cosine, which is 3 in this example.

Here are a few key points about amplitude:

• Amplitude is always a positive value.
• The cosine function oscillates between its maximum and minimum values at a distance equal to the amplitude from the equilibrium position (usually y = 0 if there's no vertical shift).
• A larger amplitude indicates a taller wave, while a smaller amplitude indicates a shorter wave.

The effect of amplitude is very visual. Imagine stretching or shrinking a sine or cosine wave vertically to reflect changes in amplitude. Understanding amplitude is crucial for analyzing the behavior of trigonometric functions in real-world applications like sound waves or tides.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. It tells us how often the pattern repeats itself. In terms of the graph, it's the distance along the x-axis before the wave pattern starts to repeat.

For a cosine function, the period can be determined by the formula \(\frac{2\pi}{B}\), where B is the coefficient of \(x\) inside the cosine term. For the function \(y = 3\cos 2x\), B equals 2, giving us a period of \(\frac{2\pi}{2} = \pi\). That means every \(\pi\) units along the x-axis, the wave pattern repeats.

• A larger B value compresses the wave, making it repeat more frequently (shorter period).
• A smaller B value stretches the wave, making it repeat less frequently (longer period).
• Standard cosine function without a coefficient (B=1) has a period of \(2\pi\).

Understanding the period of a function helps when you're analyzing oscillatory behavior in systems, like the ticking of a clock or the seasonal patterns in climate.

Cosine Function

The cosine function is one of the fundamental trigonometric functions that helps us understand wave patterns. It is defined for an angle \(\theta\), and it is periodic by nature, meaning it repeats itself in regular intervals.

Key characteristics of the cosine function include:

• It originates from the unit circle, where the cosine of an angle is the x-coordinate of the point on the circle.
• Cosine waves start at a maximum value (if no phase shift) and oscillate between this value and its opposite.
• The basic standard equation for a cosine wave is \(y = A\cos(Bx + C) + D\), where:
• \(A\) determines the amplitude.
• \(B\) affects the period of the wave.
• \(C\) causes a horizontal shift (phase shift).
• \(D\) results in a vertical shift, moving the wave up or down.

In summary, the cosine function can be adapted to many shapes and sizes, making it extremely useful in modeling different kinds of periodic behavior, from sound and light waves to the motion of pendulums in physics.