Problem 6 - Contemporary Precalculus

Question

State the amplitude, period, and phase shift of the function. \(k(t)=\cos (2 \pi t / 3)\)

Step-by-Step Solution

Answer

Answer: The amplitude is 1, the period is 3, and there is no phase shift.

Step 1: Find the Amplitude

The amplitude is the absolute value of A. Since our function has A = 1, the amplitude is \(|1| = 1\). So the amplitude of the function is 1.

Step 2: Find the Period

The period of the function depends on the value of B. The period is given by \(\frac{2\pi}{|B|}\). In our function, B = \(2\pi / 3\). So the period is: $$\frac{2\pi}{\left|\frac{2\pi}{3}\right|} = \frac{2\pi}{\frac{2\pi}{3}} = \frac{2\pi}{1} \cdot \frac{3}{2\pi} = 3$$ So the period of the function is 3.

Step 3: Find the Phase Shift

The phase shift is given by C. In our function, C = 0. As there is no phase shift, this means the function is not translated horizontally. To summarize, the amplitude of the function \(k(t) = \cos(2 \pi t / 3)\) is 1, the period is 3, and there is no phase shift.

Key Concepts

Amplitude

In trigonometric functions, amplitude refers to the height of the wave, or more specifically, how much the function oscillates above and below its central axis. It measures how far the peaks and troughs of the wave reach from this central line. Amplitude is always a positive number because it represents a distance.In the function given, \( k(t) = \cos(2 \pi t / 3) \), the amplitude is determined by the coefficient in front of the cosine function. Since there is no number explicitly multiplying the cosine function, it's understood to be 1, as in \( 1 \times \cos(2 \pi t / 3) \).A few quick points to remember about amplitude:

• The amplitude reflects the maximum height of the wave from the center.
• It is always the absolute value, making it positive.
• For \( \cos(2 \pi t / 3) \), the amplitude is \(|1| = 1\).

Understanding amplitude helps in predicting how tall or flat the waves in your trigonometric graphs will appear.

Period

The period of a trigonometric function tells us how long it takes for the function to repeat itself. It gives insight into the distance between consecutive peaks or troughs or how long one full cycle of the wave is. Knowing the period is essential for plotting the graph accurately.In the equation \( k(t) = \cos(2 \pi t / 3) \), the period is calculated by using the formula \( \frac{2\pi}{|B|} \), where \( B \) is the coefficient of \( t \) inside the cosine function. Here, \( B = \frac{2\pi}{3} \).Calculate the period:

• Substitute \( B \) into the formula: \( \frac{2\pi}{\left|\frac{2\pi}{3}\right|} \).
• Perform the division: \( \frac{2\pi}{\frac{2\pi}{3}} = 3 \).

Therefore, the period of this function is 3 units. This means the wave repeats itself every 3 units along the horizontal axis.

Phase Shift

Phase shift refers to the horizontal movement of the graph of a trigonometric function. It tells us where the function starts relative to the usual starting point. A phase shift occurs when there's a constant added or subtracted inside the function's angle.In the function \( k(t) = \cos(2 \pi t / 3) \), there isn't any additional constant added to \( t \). Normally, we'd express the cosine function as \( \cos(Bt + C) \), where \( C \) determines the phase shift. In this equation, we observe:

• The expression \( 2\pi t/3 \) doesn't show a constant \( C \), indicating \( C = 0 \).
• No addition or subtraction from \( t \) inside the cosine implies there's no horizontal shift of the graph.

This means there's no phase shift for this function, and its wave pattern starts from the usual position. Recognizing this is crucial when predicting the behavior and plotting the wave on a graph.