Problem 3 - Algebra 2

Question

Find the amplitude, if it exists, and period of each function. Then graph each function. $$ y=\frac{2}{3} \cos \theta $$

Step-by-Step Solution

Answer

Amplitude is \(\frac{2}{3}\); period is \(2\pi\).

Step 1: Understanding the Cosine Function

The function given is in the form of the basic cosine function, which is usually represented as \( y = a \cos(b\theta) \). In this function, the coefficient \( a \) affects the amplitude and \( b \) affects the period of the cosine wave.

Step 2: Identifying the Amplitude

The amplitude of \( y = \frac{2}{3} \cos \theta \) can be found by taking the absolute value of the coefficient \( a \). Here, \( a = \frac{2}{3} \). Thus, the amplitude is \( \left| \frac{2}{3} \right| = \frac{2}{3} \).

Step 3: Calculating the Period

The period of \( y = a \cos(b\theta) \) is calculated by the formula \( \frac{2\pi}{b} \). For our function \( y = \frac{2}{3} \cos \theta \), \( b = 1 \), hence the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Graphing the Function

The graph of \( y = \frac{2}{3} \cos \theta \) will look like a standard cosine wave, but the peaks will be at \( \frac{2}{3} \) and the troughs at \( -\frac{2}{3} \), instead of 1 and -1. The complete wave will repeat every \( 2\pi \) radians.

Key Concepts

Amplitude

The amplitude of a trigonometric function tells us how far the peaks and valleys of the wave stretch from the central axis. For a cosine function expressed as \( y = a \cos(b\theta) \), the amplitude is determined by the absolute value of the coefficient \( a \).

In our example, the function is \( y = \frac{2}{3} \cos(\theta) \). Here, the coefficient \( a \) is \( \frac{2}{3} \). By taking its absolute value, we find the amplitude to be \( \left| \frac{2}{3} \right| = \frac{2}{3} \).

This means that no matter what angle \( \theta \) you substitute into the cosine function, the output will never be larger than \( \frac{2}{3} \) or smaller than \(-\frac{2}{3}\). This amplitude indicates that the wave's highest point reaches \( \frac{2}{3} \) above the midline, and its lowest point dips \( \frac{2}{3} \) below.

• The amplitude is always positive, as it represents a distance and not a direction.
• It provides insight into the "strength" or "intensity" of the wave's fluctuations.

Period

The period of a trigonometric function is the distance along the horizontal axis that it takes for the function to start repeating itself. For functions in the form \( y = a \cos(b\theta) \), the period is calculated using \( \frac{2\pi}{b} \).

In the given function \( y = \frac{2}{3} \cos(\theta) \), the coefficient \( b \) is simply 1. Thus, the calculation becomes \( \frac{2\pi}{1} = 2\pi \).

This means the wave completes one full cycle, returning to its starting point, every \( 2\pi \) radians. This property hints at how "stretched" or "compressed" the wave is along the horizontal axis.

• A larger \( b \) value results in a shorter period, making the wave more "squished" together.
• A smaller \( b \) means a longer period, allowing more room for the wave to "spread out".

Understanding the period helps us comprehend how quickly the cosine wave oscillates and repeats itself.

Cosine Function

The cosine function is one of the essential trigonometric functions that arises frequently in mathematics, especially in relation to waves and oscillations. In its basic form, it is expressed as \( y = \cos(\theta) \), but it can take various transformed forms, like \( y = a \cos(b\theta + c) + d \).

For the function \( y = \frac{2}{3} \cos(\theta) \), we are specifically looking at a transformation where only the amplitude is modified from the base cosine wave.

Key aspects of the cosine function include:

• Shape: It starts at a maximum value, decreases to a minimum, and returns to the maximum—creating a smooth, continuous wave.
• Symmetry: The cosine function is symmetric about the vertical axis, meaning it is even. This results in a graph that mirrors about the \( y \)-axis.
• Applications: The cosine function is extensively used in diverse fields such as physics for wave motion, engineering, and signal processing.

Understanding the cosine function opens doors to exploring more complex phenomena involving harmonic motion and periodicity.