Problem 16 - Thomas Calculus

Question

Graph the functions in Exercises \(13-22 .\) What is the period of each function? $$ \cos \frac{\pi x}{2} $$

Step-by-Step Solution

Answer

The period of \( \cos \left( \frac{\pi x}{2} \right) \) is 4.

Step 1: Identify the Standard Form of the Cosine Function

Any cosine function can be written in the standard form \( y = a \cos(bx + c) + d \). In this exercise, the given function is \( \cos \left( \frac{\pi x}{2} \right) \). Here, we can identify \( a = 1 \), \( b = \frac{\pi}{2} \), \( c = 0 \), and \( d = 0 \).

Step 2: Determine the Period of the Function

The period of a cosine function \( y = \cos(bx) \) is given by \( \frac{2\pi}{b} \). Substitute \( b = \frac{\pi}{2} \) into the formula: \( \text{Period} = \frac{2\pi}{\frac{\pi}{2}} \).

Step 3: Simplify to Find the Period

Simplify the expression \( \frac{2\pi}{\frac{\pi}{2}} \) by multiplying by the reciprocal: \( \frac{2\pi}{\frac{\pi}{2}} = 2\pi \times \frac{2}{\pi} = 4 \). Therefore, the period of the function is 4.

Step 4: Graph the Function

To graph \( y = \cos \left( \frac{\pi x}{2} \right) \), note that the period is 4. The function will complete one full cycle over the interval \([0, 4]\). Plot key points: maximum (1) at \( x = 0 \), minimum (-1) at \( x = 2 \), and back to maximum (1) at \( x = 4 \). Repeat this for additional cycles if needed.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often written as \( y = \cos(x) \). It's part of the trio that includes sine and tangent, each relevant to angle and side relationships in right triangles. In the standard form \( y = a \cos(bx + c) + d \),

• \(a\) represents the amplitude, changing the height of the wave.
• \(b\) affects the cycle's frequency and period.
• \(c\) shifts the wave horizontally.
• \(d\) moves the wave up or down vertically.

The basic cosine wave starts at its maximum point at \(x = 0\) and oscillates between 1 and -1. For the function \(y = \cos(\frac{\pi x}{2})\), we identified that \(a = 1\), \(b = \frac{\pi}{2}\), and both \(c\) and \(d\) equal zero. This implies no amplitude scaling, no phase shift, and no vertical translation, retaining the fundamental properties of cosine except for the period adjustment.

Period of Trigonometric Functions

The period of a trigonometric function like cosine is the length of the interval over which the function completes one full cycle. For a regular cosine function \( \cos(x) \), the period is \(2\pi\). This means the function repeats every \(2\pi\) units along the x-axis.

The formula for finding the period of a cosine function in the form \( y = a \cos(bx) \) is \( \frac{2\pi}{b} \). In our exercise, the function \( y = \cos\left(\frac{\pi x}{2}\right) \) has \( b = \frac{\pi}{2} \). By substituting this value into the formula, we find the period:

• Calculate: \( \frac{2\pi}{\frac{\pi}{2}} = 4 \).

Thus, the period of \( \cos\left(\frac{\pi x}{2}\right) \) is 4. This means that the cosine wave will repeat every 4 units along the x-axis, demonstrating how manipulating \(b\) compresses or stretches the wave horizontally.

Graphing Trigonometric Functions

Graphing trigonometric functions can be simplified by understanding their key characteristics. For the cosine function, these include its amplitude, period, and key points.

To graph \( y = \cos\left(\frac{\pi x}{2}\right) \), follow these steps:

• Identify the period as 4.
• The maximum value (1) occurs at the starting point \( x = 0 \).
• The minimum value (-1) occurs halfway through the period at \( x = 2 \).
• Return to maximum at the end of the period \( x = 4 \).

Throughout the interval \([0, 4]\), plot these key points and sketch the wave smoothly moving through these values. Extending the graph beyond \([0, 4]\), replicate this cycle to cover more of the x-axis as needed. Understanding this cyclical behavior assists in predicting the cosine function's values across its domain.