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The period of a function is an essential aspect to understand when studying trigonometric functions. A function's period represents the interval of the x-axis that completes one full cycle of the wave pattern before it starts repeating again. For trigonometric functions like the sine and cosine, the standard period is typically given by the length for which the function returns to its starting value along the y-axis.
The general formula to find the period of a sine function, such as \(y = rac{1}{5} \, \sin 6x\), involves the coefficient of \(x\) in the function. If we have a function in the form \(y = a \, \sin(bx)\), then we use the formula \(\frac{2\pi}{|b|}\) to determine its period. In this specific problem, the coefficient \(b\) is 6, so the period calculation is \(\frac{2\pi}{|6|} = \frac{\pi}{3}\).
This means the sine wave completes one full cycle every \(\frac{\pi}{3}\) units along the x-axis. Recognizing the period helps in graphing the function accurately and predicting its behavior further along the axis.

Amplitude refers to the height of the wave-like pattern formed by a trigonometric function, measured from the central axis (or the horizontal midline) to the peak of the wave. In mathematical terms, for a sine function like \(y = a \, \sin(bx)\), the amplitude is given by the absolute value of \(a\).
In the function \(y = \frac{1}{5} \, \sin 6x\), the amplitude can be found by taking the absolute value of \(\frac{1}{5}\), which is \(\frac{1}{5}\) or 0.2. This implies that the maximum vertical height of the wave above or below the central axis is 0.2 units.
The amplitude affects how "tall" or "short" the wave appears on a graph, and understanding this helps in accurately drawing and interpreting the sine or cosine functions.

The sine function is one of the fundamental trigonometric functions in mathematics, characterized by its periodic wave-like shape. It is defined as \(y = a \, \sin(bx)\), where \(a\) and \(b\) are constants that affect the function's properties.
The sine wave repeatedly oscillates between -1 and 1 under normal conditions, which reflects periodic behavior. Adjustments to the equation parameters \(a\) and \(b\) alter the function's amplitude and period, respectively.
Understanding how to manipulate these parameters in \(y = \frac{1}{5} \, \sin 6x\) provides insights into its graphical representation:

• The coefficient \(a = \frac{1}{5}\) changes the amplitude, making the peaks of the wave lower.
• The coefficient \(b = 6\) compresses the period, resulting in cycles that are nearer together.

This understanding is foundational for students learning trigonometry as it ties into solving equations and predicting function behavior.