Problem 89 - Algebra 2

Question

Write a cosine function for each description. amplitude \(3,\) period 2\(\pi\)

Step-by-Step Solution

Answer

The cosine function corresponding with an amplitude of 3 and a period of 2\(\pi\) is \(y = 3 \cos(x)\).

Step 1: Identify the amplitude

Amplitude is given as 3. Verifying this, we see it matches the absolute value of A in the function \(y = A \cos(Bx)\). Therefore, A = 3.

Step 2: Identify the period

Period is given as 2\(\pi\). Checking this, it corresponds with the application of the formula \(B = \frac{2 \pi}{P}\). Substituting P = 2\(\pi\) to find B, we get B = 1.

Step 3: Substitute A and B into the function

Substituting A = 3 and B = 1 into the function \(y = A \cos(Bx)\), the cosine function is therefore \(y = 3 \cos(x)\).

Key Concepts

Amplitude

The amplitude of a trigonometric function is one of its defining characteristics. Think of it as the function's "height." Specifically, the amplitude determines how far the graph rises above and falls below the midline. This is the vertical distance from the middle of the wave to its peak or trough.
For a cosine function, which can be written as \(y = A \cos(Bx)\), the amplitude is represented by the absolute value of \(A\). So, when given an amplitude of 3, it means the graph of the cosine function will rise to 3 units above and dip 3 units below its central axis.
Understanding amplitude is crucial because it affects the shape and appearance of the graph. It's important to remember:

• Amplitude is always positive and is denoted as \(|A|\).
• It does not affect the horizontal position or the period of the function.

In context with this exercise, the amplitude is clearly set at 3, which guides the formation of the sine wave's vertical stretch.

Period

The period of a trigonometric function indicates the distance over which the function completes one full cycle. Essentially, it's how long it takes for the function to repeat its pattern.
In mathematical terms, the period \(P\) of a cosine function \(y = A \cos(Bx)\) can be determined using the formula \(P = \frac{2 \pi}{B}\). This equation showcases the relationship between the frequency factor \(B\) and the period length.
For the given exercise, the period is defined as \(2\pi\), matching that of a standard cosine wave. By substituting \(P = 2\pi\) into the equation \(B = \frac{2\pi}{P}\), we find \(B = 1\), indicating that the wave completes one full cycle over a span of \(2\pi\) radians.

• The smaller the period, the more frequent the waves, as it fits more cycles into the same span.
• The larger the period, the more spread out the waves, providing a more extended interval before the cycle repeats.

Grasping the concept of a period helps in predicting and analyzing the behavior of trigonometric graphs over different intervals.

Trigonometric Functions

Trigonometric functions are foundational mathematical entities used to relate the angles of a triangle to the lengths of its sides. They are crucial in fields such as physics, engineering, and astronomy.
The primary trigonometric functions include sine, cosine, and tangent. These functions are used to model periodic phenomena such as sound waves, light waves, and the motion of pendulums.
The cosine function, in particular, is often represented as \(y = A \cos(Bx) + D\), where:

• \(A\) represents the amplitude, affecting the wave's height.
• \(B\) impacts the frequency or period, influencing how often waves occur.
• \(D\) shifts the function up or down vertically.

For the exercise at hand, the function we derived was \(y = 3 \cos(x)\). This reflects the defining characteristics of cosine waves, leveraging their periodic nature to represent consistently repeating patterns.
Recognizing how these functions represent real-world scenarios empowers students to model and predict complex systems with greater accuracy.