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The concept of amplitude in trigonometric functions is crucial because it tells us how tall the waves are. Amplitude refers to the maximum distance the wave peaks reach from the middle or baseline of the wave. For a cosine or sine wave, which oscillates above and below a central line, the amplitude is simply the coefficient of the sine or cosine function.

In simple terms:

• Amplitude is the 'height' of the wave.
• It affects the vertical stretch of the graph.
• To find the amplitude, look for the number before \cos\ or \sin\ in the equation.

For example, in the function \( y = 0.2 \cos\left(\frac{2\pi}{0.8} t\right) \), the amplitude is \( 0.2 \). Similarly, in \( y = 5 \cos\left( \frac{1}{8}t + \frac{\pi}{6} \right) \), the amplitude is \( 5 \). These values show the maximum and minimum points the wave can reach on the graph, providing a simple way to visualize how far the wave stretches along the y-axis.

The period of a trigonometric function is the distance over which the function repeats its shape. To understand the period, consider how long it takes for a pattern in the graph to start over. The formula to calculate the period for sine and cosine functions is \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( t \) inside the function.

Key points about period:

• The period dictates how "wide" a cycle of the wave is on the x-axis.
• It's the length for one complete cycle of the wave.
• For a function \( f(t) = \cos(Bt) \), compute the period using: \( \text{Period} = \frac{2\pi}{B} \).

In our exercises, for the first function \( y = 0.2 \cos\left( \frac{2 \pi}{0.8} t \right) \), the period is \( 0.8 \). That means every 0.8 units, the wave begins its cycle again. For the second function \( y = 5 \cos\left( \frac{1}{8}t + \frac{\pi}{6} \right) \), the calculation gives a period of \( 16 \). This means on the graph, the wave pattern stretches over 16 units before repeating. Understanding the period helps in accurately plotting and predicting future points of the wave's cycle.

Phase shift in trigonometric functions refers to the horizontal movement of the wave along the x-axis. This shift can slide the graph left or right depending on the direction indicated by the function's equation. Simply put, the phase shift indicates where the wave starts.

Important facts about phase shift:

• Phase shift shows how far to move the graph horizontally from its standard position.
• Calculated by resolving \( Bt + C = 0 \) for \( t \) in \( f(t) = \cos(Bt + C) \).
• A positive phase shift moves the graph to the left, while a negative shift moves it to the right.

For instance, in our function \( y = 0.2 \cos\left(\frac{2\pi}{0.8} t \right) \), there is no additional term added to \( t \), so no phase shift occurs. The graph starts from its standard position. However, in \( y = 5 \cos\left( \frac{1}{8}t + \frac{\pi}{6} \right) \), the phase shift is calculated by setting the inside term to zero: \( \frac{1}{8}t + \frac{\pi}{6} = 0 \). Solving this results in a phase shift of \(-\frac{4\pi}{3}\); when converted roughly, this is a rightward shift of about 4.19 units. Phase shifts are essential for marking the wave's correct starting point on a graph.