91Ó°ÊÓ

The sine function is a fundamental trigonometric function that appears frequently in periodic phenomena. It is defined for all real numbers and oscillates between -1 and 1. In the context of the seasonal sales function given by \[ S(t) = 2000 + 600 \sin \left( \frac{\pi}{6} t \right) \]this sine component, \( \sin \left( \frac{\pi}{6} t \right) \), creates the oscillation that reflects seasonal variations in sales. The factor \( \frac{\pi}{6} \) influences the frequency of the sine wave, ensuring that one full cycle occurs over 12 months. This periodic behavior is typical for businesses affected by seasons, with ups and downs happening in a regular pattern.These oscillations are superimposed on a steady base sales level of 2000 units. The amplitude of 600 represents the magnitude of fluctuation above and below this base level. Understanding the characteristics of sine functions helps us model such periodic trends effectively.

In trigonometric functions, identifying the maximum and minimum values is crucial for understanding their behavior. These values are essential in determining the peaks and troughs of the seasonal sales.For the sales function:- The maximum value occurs when the sine reaches 1: \[ S(t) = 2000 + 600 \times 1 = 2600. \] This happens at \( t = 3 \) months, which corresponds to April 1st. This indicates that April experiences the highest sales.- The minimum value occurs when the sine reaches -1: \[ S(t) = 2000 - 600 \times 1 = 1400. \] This occurs at \( t = 9 \) months or October 1st, denoting the month with the lowest sales.By identifying these points, businesses can strategize for inventory, staffing, and marketing to maximize customer engagement and sales throughout the year.

Derivatives provide insight into the rate of change of a function, which is particularly useful for understanding how fast something is increasing or decreasing. For the seasonal sales function, the derivative is found by differentiating the main function:\[ S'(t) = 100\pi \cos \left(\frac{\pi}{6} t \right). \]By substituting \( t = 2 \), we find:\[ S'(2) = 100\pi \cos \left(\frac{\pi}{3} \right) = 100\pi \times \frac{1}{2} = 50\pi \approx 157.1. \]This value represents the rate of change in sales at \( t = 2 \) months, meaning the sales are increasing at a rate of approximately 157.1 units per month. Derivative interpretation is vital for businesses to gauge how swiftly sales are growing or declining at any given moment, allowing them to make informed decisions about marketing strategies and inventory management.

Graphing periodic functions such as sine waves is a valuable tool for visualizing complex behaviors in data. For the given function, plotting \[ S(t) = 2000 + 600 \sin \left( \frac{\pi}{6} t \right) \]from \( t = 0 \) to \( t = 12 \) illustrates the trend over one year. A graph helps to clearly display the repeating pattern, showing peaks at January and July, and troughs at April and October. To graph:- Identify critical points: maximum, minimum, and midpoints where the sine function is zero.- Plot these points to see the shape of the curve.- Note that the amplitude and the vertical shift (2000) sets the baseline and range of the graph.The graph makes it easier for stakeholders to predict future sales trends and plan accordingly. By setting the graph within a real-world context, like monthly sales data, it becomes a practical tool for understanding and reacting to cyclical swings in business performance.