Finding The Period Of The Cotangent Function Y = (3/2)cot((3/5)x) + 5

In the realm of trigonometry, understanding the periodic nature of functions is crucial for analyzing their behavior and applications. Among these functions, the cotangent function stands out with its unique characteristics. This article delves into the process of determining the period of a cotangent function, specifically focusing on the function $\bf{y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5}$. We will explore the fundamental concepts, apply relevant formulas, and provide a step-by-step solution to find the period of the given function. Moreover, we will discuss the significance of understanding the period in various contexts and highlight the key properties of cotangent functions.

Understanding Periodic Functions

Periodic functions are the cornerstone of many mathematical and scientific models, describing phenomena that repeat over regular intervals. A function $f(x)$ is said to be periodic if there exists a non-zero constant $P$ such that $f(x + P) = f(x)$ for all $x$ in the domain of $f$. The smallest such positive constant $P$ is called the period of the function. In simpler terms, the period is the length of one complete cycle of the function's graph.

The Significance of Periodicity

Periodicity is a fundamental concept that arises in numerous real-world applications. From the cyclical patterns of planetary orbits to the oscillations of electrical circuits and the rhythmic nature of sound waves, periodic functions provide a powerful tool for modeling and understanding these phenomena. In mathematics, understanding the period of a function is essential for graphing, analyzing, and solving equations involving trigonometric functions.

Key Examples of Periodic Functions

Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, are prime examples of periodic functions. Each of these functions exhibits a unique periodic behavior, characterized by its specific period and amplitude. Sine and cosine functions have a period of $2\pi$, while tangent and cotangent functions have a period of $\pi$. These fundamental periods are crucial in determining the periods of more complex trigonometric functions.

Exploring the Cotangent Function

The cotangent function, denoted as $\cot(x)$, is one of the six fundamental trigonometric functions. It is defined as the ratio of the cosine to the sine of an angle: $\cot(x) = \frac{\cos(x)}{\sin(x)}$. The cotangent function is the reciprocal of the tangent function, meaning $\cot(x) = \frac{1}{\tan(x)}$. Understanding the properties of the cotangent function is essential for determining its period and analyzing its behavior.

Properties of the Cotangent Function

• Periodicity: The cotangent function is periodic with a period of $\pi$. This means that $\cot(x + \pi) = \cot(x)$ for all $x$ in its domain. The period of $\pi$ is a fundamental characteristic of the cotangent function and is crucial for understanding its behavior.
• Vertical Asymptotes: The cotangent function has vertical asymptotes at integer multiples of $\pi$, i.e., at $x = n\pi$, where $n$ is an integer. This is because the sine function, which is in the denominator of the cotangent function, is zero at these points, making the cotangent function undefined.
• Zeros: The cotangent function has zeros at odd multiples of $\frac{\pi}{2}$, i.e., at $x = \frac{(2n + 1)\pi}{2}$, where $n$ is an integer. At these points, the cosine function is zero, making the cotangent function zero as well.
• Symmetry: The cotangent function is an odd function, meaning that $\cot(-x) = -\cot(x)$. This symmetry property can be helpful in analyzing the graph of the cotangent function and understanding its behavior over different intervals.

Transformations of the Cotangent Function

The basic cotangent function, $\cot(x)$, can undergo various transformations, such as vertical and horizontal stretches, compressions, and shifts. These transformations affect the period, amplitude, and position of the graph. Understanding how these transformations impact the cotangent function is essential for analyzing and graphing more complex cotangent functions.

Determining the Period of $y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5$

Now, let's focus on the specific function given: $\bf{y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5}$. To find the period of this function, we need to consider the transformations applied to the basic cotangent function, $\cot(x)$. The general form of a transformed cotangent function is:

$y = A \cot(Bx - C) + D$

where:

• $A$ is the vertical stretch or compression factor.
• $B$ affects the period of the function.
• $C$ is the horizontal shift.
• $D$ is the vertical shift.

The Role of the Coefficient B

The coefficient $B$ plays a crucial role in determining the period of the transformed cotangent function. The period of the transformed function is given by:

$\text{Period} = \frac{\pi}{|B|}$

This formula highlights the inverse relationship between the coefficient $B$ and the period of the function. A larger value of $|B|$ results in a shorter period, while a smaller value of $|B|$ results in a longer period.

Applying the Formula to Our Function

In our function, $y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5$, we can identify the coefficients as follows:

• $A = \frac{3}{2}$
• $B = \frac{3}{5}$
• $C = 0$
• $D = 5$

To find the period, we use the formula:

$\text{Period} = \frac{\pi}{|B|} = \frac{\pi}{|\frac{3}{5}|} = \frac{\pi}{\frac{3}{5}} = \frac{5\pi}{3}$

Therefore, the period of the function $y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5$ is $\frac{5\pi}{3}$ units.

Step-by-Step Solution

Let's summarize the steps we took to find the period of the function:

1. Identify the coefficient B: In the given function, $y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5$, the coefficient $B$ is $\frac{3}{5}$.
2. Apply the period formula: The period of a transformed cotangent function is given by $\text{Period} = \frac{\pi}{|B|}$.
3. Substitute the value of B: Substitute $B = \frac{3}{5}$ into the formula: $\text{Period} = \frac{\pi}{|\frac{3}{5}|}$.
4. Simplify the expression: Simplify the expression to find the period: $\text{Period} = \frac{\pi}{\frac{3}{5}} = \frac{5\pi}{3}$.

Thus, the period of the function $y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5$ is $\frac{5\pi}{3}$ units.

Graphical Interpretation of the Period

The period of a function is visually represented on its graph as the distance along the x-axis over which the function completes one full cycle. For the function $y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5$, the graph will repeat itself every $\frac{5\pi}{3}$ units along the x-axis. This means that if you were to trace the graph of the function, after every interval of $\frac{5\pi}{3}$, the pattern would start repeating itself.

Visualizing the Period

Imagine the graph of the cotangent function stretched and compressed. The coefficient $\frac{3}{5}$ inside the cotangent function compresses the graph horizontally, effectively shortening the period. The factor $\frac{3}{2}$ vertically stretches the graph, while the $+5$ shifts the entire graph upwards by 5 units. However, only the horizontal compression affects the period.

To visualize the period, you could graph the function using a graphing calculator or software. You would observe that the function repeats its pattern over intervals of length $\frac{5\pi}{3}$. The vertical asymptotes, which are normally at integer multiples of $\pi$, are now compressed and shifted due to the $\frac{3}{5}x$ term inside the cotangent.

Practical Implications of the Period

Understanding the period of a trigonometric function is crucial in various applications. For instance, in physics, the period of a wave (such as a sound wave or an electromagnetic wave) determines its frequency. In engineering, the period of a periodic signal is essential for designing and analyzing systems that process these signals. In mathematics, knowing the period of a function allows us to predict its behavior over extended intervals and solve equations involving the function.

Importance of Understanding the Period

The period of a function is a fundamental concept in mathematics, particularly in the study of trigonometric functions. It provides valuable information about the function's cyclical behavior and helps in analyzing its graph and properties. A clear understanding of the period is crucial for solving problems involving trigonometric functions and for applying these functions in various fields.

Applications in Mathematics and Beyond

In mathematics, the period is essential for graphing trigonometric functions, solving trigonometric equations, and analyzing the behavior of periodic phenomena. Beyond mathematics, the concept of periodicity finds applications in various fields, including physics, engineering, and signal processing. For example, in physics, the period of a pendulum's swing or the oscillations of a spring-mass system is a critical parameter. In engineering, the period of an alternating current (AC) signal is fundamental to electrical circuit design. In signal processing, the period of a signal is used to analyze its frequency components.

Common Mistakes to Avoid

When determining the period of a transformed trigonometric function, it's important to avoid common mistakes. One common mistake is to overlook the impact of the coefficient $B$ on the period. Remember that the period is inversely proportional to the absolute value of $B$. Another mistake is to confuse the period with other parameters, such as the amplitude or phase shift. It's crucial to focus on the specific formula for the period and apply it correctly.

Conclusion

In conclusion, determining the period of the function $y = \frac{3}{2} \cot \left(\frac{3}{5} x\right) + 5$ involves understanding the properties of the cotangent function and applying the appropriate formula. By identifying the coefficient $B$ and using the formula $\text{Period} = \frac{\pi}{|B|}$, we found that the period of the function is $\frac{5\pi}{3}$ units. This process highlights the importance of understanding the transformations applied to trigonometric functions and their impact on the period. Mastering the concept of periodicity is essential for analyzing and applying trigonometric functions in various mathematical and real-world contexts. The ability to accurately determine the period of a trigonometric function is a valuable skill for students and professionals alike.