Problem 88 - Algebra 2

Question

Write a cosine function for each description. amplitude \(4,\) period 8

Step-by-Step Solution

Answer

The cosine function for the given description is \(y = 4\cos(\pi/4x)\).

Step 1: Understanding amplitude and period

The amplitude of the cosine function is the value given directly in front of the cosine keyword. The period is inversely proportional to the absolute value of B, where B is the coefficient of x inside the cosine function. The formula to get the period is \(2\pi/|B|\). Therefore, to find the value of B we should use the formula \(B = 2\pi / \text{period}\).

Step 2: Finding the value of B

Given that the period is 8, the formula gives us \(B = 2\pi / 8 = \pi/4\). This means the function repeats every 8 units.

Step 3: Writing the cosine function

Finally, substitute the given amplitude and calculated B into the general cosine function form \(y = A\cos(Bx)\). The cosine function is \(y = 4\cos(\pi/4x)\).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are essential in studying periodic phenomena such as sound waves, light waves, and other periodic motion. The most common trigonometric functions include sine, cosine, and tangent. In trigonometry, the cosine function, often denoted as \( \cos \), is particularly significant because it describes the east-west component of a unit circle. This function maps any angle to a value between -1 and 1.

• Trigonometric functions are periodic, meaning they repeat at regular intervals.
• The cosine function is even, which means that \( \cos(-x) = \cos(x) \).
• They are used to model cyclic phenomena like tides, seasons, and musical notes.

When given a trigonometric equation, it's important to understand how factors such as amplitude and period affect the graph's shape. In this context, understanding the basics of trigonometric functions helps in exploring how these parameters transform a basic cosine wave into one that fits particular criteria.

Exploring Amplitude

Amplitude is a key feature of the cosine function that affects how 'tall' or 'short' the wave appears on a graph. It describes the maximum extent of oscillation from the central axis. Amplitude is represented by the coefficient in front of the cosine function, denoted generally as \( A \).

• The amplitude affects only the vertical stretching or shrinking of the graph.
• If the amplitude \( A \) is positive, the cosine graph retains its standard orientation.
• If \( A \) is negative, the graph undergoes a vertical reflection over the x-axis.

In the exercise, the amplitude is given as 4. This means the graph stretches vertically from -4 to 4, maintaining the regular oscillation pattern of cosine along these limits. Knowing the amplitude also helps to quickly identify how intense or subdued the wave motion will be visually.

Determining the Period

Period is another fundamental aspect of the cosine function which determines the length of one complete cycle of the wave before it begins to repeat. It is calculated using the formula \( \text{Period} = \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( x \) in the cosine function.

• A larger period implies the wave stretches out more and takes longer to complete a cycle.
• A smaller \( B \) increases the period, thereby lengthening the cycle.

In the given exercise, the period is specified as 8, leading to the calculation of \( B \) as \( \frac{\pi}{4} \). This means the cosine wave completes a full cycle over 8 units on the x-axis, reflecting a slower frequency compared to the standard cycle length \( 2\pi \). Understanding the period is crucial for determining how frequently the graph repeats its pattern, essential for applications involving time-dependent growth and oscillation modeling.