91Ó°ÊÓ

In trigonometry, the amplitude is a critical concept for understanding wave functions. It refers to the peak value or the maximum extent of the wave from its equilibrium position. Essentially, amplitude tells you how "tall" or "short" each wave peak is.When dealing with the sine function, the amplitude is the absolute value of the coefficient that appears in front of the sine term. For the function provided, which is \( f(x) = -\sin x \), the coefficient in front of \( \sin x \) is \(-1\).Since amplitude is always a positive number, we take the absolute value:

• The amplitude is calculated as \(|-1| = 1\).

This means that the sine wave oscillates between 1 and -1, reflecting from the horizontal axis with these maximum and minimum values.

The period of a trigonometric function is an important aspect as it determines how often the wave repeats itself. For a regular sine wave, this period is the length of one complete cycle, after which the pattern repeats.To find the period of a sine function of the form \( \sin(bx) \), you use the formula:

• Period \( = \frac{2\pi}{b} \).

In the given function \( f(x) = -\sin x \), \( b = 1 \), so the period is:

• \( \frac{2\pi}{1} = 2\pi \).

This means the function completes one full cycle every \(2\pi\) units along the x-axis. Understanding this helps in predicting and graphing the sine waves as they move along the x-axis.

Horizontal shift, also known as phase shift, describes the sideways movement of a wave along the x-axis. It alters the starting point of the wave, either shifting it left or right.Typically, a sine function written as \( \sin(x - c) \) has its horizontal shift determined by the constant \( c \), where:

• If \( c > 0 \), the function shifts to the right.
• If \( c < 0 \), the function shifts to the left.

For the sine function \( f(x) = -\sin x \), we note that there is no horizontal shift because there is no \( c \) value shifting the function left or right. So:

• Horizontal shift \( = 0 \).

This implies that the function begins its cycle at the origin without any lateral displacement from its starting point.

In the context of trigonometric functions, the average value refers to the mean value of the function over one complete cycle. It's crucial for understanding the behavior of the function over its period.For sinusoidal functions such as sine and cosine, the average value is affected by any vertical shifts in the graph. Typically, if the graph is not shifted vertically, the average value remains 0. Given the function \( f(x) = -\sin x \), there is no vertical displacement. This results in:

• The average value over a full cycle remains 0.

Understanding the average value is essential as it provides a baseline that represents the middle, or the neutral level, of the wave movement for the sine function over time.