What Is The Period Of The Trigonometric Equation F(x)=-8sin((5π/3)x+π/6)+4? Express Your Answer As A Simplified Fraction.

In mathematics, trigonometric functions play a crucial role, particularly in modeling periodic phenomena. Understanding the period of a trigonometric function is essential for analyzing its behavior and applications. This article delves into the concept of the period of a trigonometric equation, providing a step-by-step guide on how to determine it. We will use the example function $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$ to illustrate the process. Let's embark on this mathematical journey to unravel the intricacies of trigonometric periods.

Understanding Trigonometric Functions and Their Periods

Trigonometric functions are fundamental in mathematics, particularly in areas like calculus, physics, and engineering. These functions, including sine, cosine, tangent, cotangent, secant, and cosecant, describe the relationships between angles and sides of triangles. Their periodic nature makes them invaluable for modeling cyclical phenomena such as oscillations, waves, and rotations. When we talk about the period of a trigonometric function, we refer to the interval over which the function's graph completes one full cycle before repeating itself. Identifying the period is crucial for predicting the function's behavior over time or space.

The sine and cosine functions, denoted as sin(x) and cos(x), respectively, are the cornerstone of trigonometric functions. Their basic form, y = A sin(Bx - C) + D or y = A cos(Bx - C) + D, involves several parameters that influence their graphs. The parameter A represents the amplitude, which determines the vertical stretch of the function. The parameter B affects the period, the horizontal length of one complete cycle. The parameter C introduces a phase shift, shifting the graph horizontally. Lastly, the parameter D signifies the vertical shift, moving the graph up or down. Understanding how each parameter influences the graph is key to analyzing trigonometric functions. For instance, a larger amplitude (A) means the function will have a greater vertical range, while a change in B will compress or stretch the graph horizontally, thereby altering the period. Similarly, the phase shift (C) will move the graph left or right, and the vertical shift (D) will raise or lower the entire graph. These transformations allow trigonometric functions to model a wide range of periodic phenomena accurately. Recognizing the role of each parameter is vital for manipulating and interpreting trigonometric functions effectively.

Determining the Period of $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$

To find the period of the given trigonometric function $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$, we need to focus on the coefficient of $x$ inside the sine function. The general form of a sine function is $f(x) = A ext{sin}(Bx - C) + D$, where A is the amplitude, B is related to the period, C is the phase shift, and D is the vertical shift. In our case, $B = \frac{5\pi}{3}$. The period T of a sine function is given by the formula:

$T = \frac{2\pi}{|B|}$

Here, $|B|$ denotes the absolute value of B. This formula arises from the fact that the standard sine function, sin(x), has a period of $2\pi$. When we multiply x by B, we effectively compress or stretch the graph horizontally, thereby changing the period. To find the new period, we divide the original period ($2\pi$) by the absolute value of B. This formula is the cornerstone for calculating the period of any sinusoidal function. Applying this to our given function, we substitute $B = \frac{5\pi}{3}$ into the formula:

$T = \frac{2\pi}{|\frac{5\pi}{3}|}$

Since $\frac{5\pi}{3}$ is already positive, we can drop the absolute value signs:

$T = \frac{2\pi}{\frac{5\pi}{3}}$

To simplify this fraction, we multiply the numerator by the reciprocal of the denominator:

$T = 2\pi \cdot \frac{3}{5\pi}$

The $\pi$ terms cancel out, leaving us with:

$T = \frac{2 \cdot 3}{5}$

Thus, the period T is:

$T = \frac{6}{5}$

Therefore, the period of the trigonometric function $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$ is $\frac{6}{5}$. This means the function completes one full cycle over an interval of $\frac{6}{5}$ units along the x-axis. Understanding this period allows us to predict the function's behavior over any given interval. The coefficient of x within the trigonometric function dictates how frequently the function oscillates, and the formula $T = \frac{2\pi}{|B|}$ provides a direct way to quantify this oscillation in terms of the period.

Step-by-Step Calculation of the Period

To solidify the process of determining the period of a trigonometric function, let's reiterate the step-by-step calculation for $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$. This methodical approach ensures accuracy and clarity in understanding the underlying principles. The first critical step involves identifying the coefficient of x inside the trigonometric function. In our given function, $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$, the coefficient of x is $\frac{5\pi}{3}$. This value, often denoted as B in the general form of a trigonometric function $f(x) = A ext{sin}(Bx - C) + D$, plays a pivotal role in determining the period.

The second step is to apply the period formula. The period T of a sine function is calculated using the formula $T = \frac{2\pi}{|B|}$, where $|B|$ represents the absolute value of B. This formula stems from the fundamental property that the standard sine function, sin(x), has a period of $2\pi$. The coefficient B scales the x-axis, either compressing or stretching the graph horizontally, thereby altering the period. The absolute value ensures that the period is always a positive quantity, representing the length of one complete cycle. Substituting the value of B from our function into the formula, we get:

$T = \frac{2\pi}{|\frac{5\pi}{3}|}$

The third step is to simplify the expression. Since $\frac{5\pi}{3}$ is a positive number, its absolute value is simply $\frac{5\pi}{3}$. Therefore, the equation becomes:

$T = \frac{2\pi}{\frac{5\pi}{3}}$

To divide by a fraction, we multiply by its reciprocal. Thus, we rewrite the equation as:

$T = 2\pi \cdot \frac{3}{5\pi}$

Notice that the $\pi$ terms appear in both the numerator and the denominator, allowing us to cancel them out. This simplification leads to:

$T = \frac{2 \cdot 3}{5}$

Performing the multiplication, we arrive at the final result:

$T = \frac{6}{5}$

This result signifies that the function $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$ completes one full cycle over an interval of $\frac{6}{5}$ units along the x-axis. This step-by-step approach provides a clear and concise method for calculating the period of any trigonometric function. Understanding and applying these steps are crucial for analyzing and predicting the behavior of trigonometric functions in various mathematical and real-world contexts.

Practical Applications of Understanding the Period

Understanding the period of trigonometric functions has numerous practical applications across various fields. Trigonometric functions, due to their periodic nature, are invaluable tools for modeling phenomena that repeat over time or space. One significant application is in physics, where trigonometric functions are used to describe oscillations and waves. For instance, the motion of a pendulum, the vibration of a guitar string, and the propagation of electromagnetic waves can all be modeled using sinusoidal functions. The period of these functions directly corresponds to the time it takes for one complete cycle of the oscillation or wave. Knowing the period allows physicists to predict the behavior of these systems over time, making it possible to design and analyze various physical systems and devices.

Another key area where the period of trigonometric functions is crucial is in electrical engineering. Alternating current (AC) circuits, which are fundamental to modern electrical systems, are characterized by sinusoidal voltages and currents. The period of these sinusoidal waveforms determines the frequency of the AC signal, which is a critical parameter in the design and operation of electrical devices. For example, the standard frequency of AC power in many countries is 50 or 60 Hz, which corresponds to a period of 20 or 16.67 milliseconds, respectively. Electrical engineers use the period and frequency to analyze circuit behavior, design filters, and ensure the stable operation of electrical grids and devices. Understanding the periodic nature of AC signals is essential for ensuring the efficient and safe transmission and utilization of electrical energy.

In the field of signal processing, trigonometric functions are used to analyze and manipulate signals, such as audio and radio waves. Fourier analysis, a powerful technique for decomposing complex signals into simpler sinusoidal components, relies heavily on the concept of periodicity. The period of the sinusoidal components provides information about the frequencies present in the signal, which is crucial for tasks such as noise reduction, signal compression, and pattern recognition. Signal processing engineers use the period to extract meaningful information from signals, enabling the development of advanced communication systems, audio processing algorithms, and medical imaging techniques.

Beyond these specific fields, the concept of periodicity and the understanding of trigonometric functions are also relevant in areas such as economics, biology, and music. Economic cycles, such as business cycles, often exhibit periodic behavior that can be modeled using trigonometric functions. Biological rhythms, such as circadian rhythms, also display periodic patterns that are essential for regulating various physiological processes. In music, the frequencies of musical notes and the rhythms of musical compositions are inherently periodic. The widespread applicability of trigonometric functions and the concept of periodicity underscores their fundamental importance in understanding and modeling the world around us. The period, as a key parameter of these functions, provides a valuable tool for analyzing and predicting the behavior of diverse systems and phenomena.

Conclusion

In conclusion, understanding the period of a trigonometric function is fundamental for analyzing its behavior and applications. By focusing on the coefficient of x and applying the formula $T = \frac{2\pi}{|B|}$, we can accurately determine the period. Using the example function $f(x) = -8 ext{sin}(\frac{5\pi}{3}x + \frac{\pi}{6}) + 4$, we demonstrated a step-by-step method to find that its period is $\frac{6}{5}$. This knowledge is not only crucial in mathematics but also in various practical fields, including physics, engineering, and signal processing, where trigonometric functions are used to model periodic phenomena. Mastering the concept of the period allows for a deeper understanding of oscillations, waves, and other cyclical behaviors, paving the way for advancements in numerous scientific and technological domains. The ability to identify and interpret the period of trigonometric functions is an essential skill for anyone working with these powerful mathematical tools.