Question

Determine the period, the sinusoidal axis, and the amplitude for each of the following. a) The first maximum of a sine function occurs at the point \(\left(30^{\circ}, 24\right),\) and the first minimum to the right of the maximum occurs at the point \(\left(80^{\circ}, 6\right).\) b) The first maximum of a cosine function occurs at \((0,4),\) and the first minimum to the right of the maximum occurs at \(\left(\frac{2 \pi}{3},-16\right).\) c) An electron oscillates back and forth 50 times per second, and the maximum and minimum values occur at +10 and \(-10,\) respectively.

Short answer

Part a: Amplitude = 9, Axis = 15, Period = 100°. Part b: Amplitude = 10, Axis = -6, Period = \(\frac{4 \pi}{3}\). Part c: Amplitude = 10, Axis = 0, Period = \(\frac{1}{50}\) seconds.

Step-by-step solution

Part a) Determine the Amplitude

The amplitude of a sine function is half the difference between the maximum and minimum values. Here, the maximum is 24 and the minimum is 6. So, amplitude = \(\frac{24 - 6}{2} = 9\).

Part a) Determine the Sinusoidal Axis

The sinusoidal axis is the average of the maximum and minimum values. Calculate: \(\frac{24 + 6}{2} = 15\).

Part a) Determine the Period

The period is twice the distance between consecutive maximum and minimum points. Here, the distance between \(30^{\circ}\) and \(80^{\circ}\) is \(80^{\circ} - 30^{\circ} = 50^{\circ}\). Therefore, the period is \(50 \times 2 = 100^{\circ}\).

Part b) Determine the Amplitude

For the cosine function, the amplitude is half the difference between the maximum and minimum values. Here, max = 4 and min = -16. So, amplitude = \(\frac{4 - (-16)}{2} = 10\).

Part b) Determine the Sinusoidal Axis

The sinusoidal axis is the average of the maximum and minimum values. Calculate: \(\frac{4 + (-16)}{2} = -6\).

Part b) Determine the Period

The period is twice the distance between consecutive maximum and minimum points. The distance between \(0\) and \(\frac{2 \pi}{3}\) is \(\frac{2 \pi}{3}\). Therefore, the period is \(2 \times \frac{2 \pi}{3} = \frac{4 \pi}{3}\).

Part c) Determine the Amplitude

The amplitude is half the difference between the maximum and minimum values. Here, max = 10 and min = -10. So, amplitude = \(\frac{10 - (-10)}{2} = 10\).

Part c) Determine the Sinusoidal Axis

The sinusoidal axis is the average of the maximum and minimum values. Calculate: \(\frac{10 + (-10)}{2} = 0\).

Part c) Determine the Period

The period is the reciprocal of the frequency. Here, the frequency is 50 times per second, so period = \(\frac{1}{50} \text{ seconds}\).

Key concepts

Amplitude Calculation

In sinusoidal functions, amplitude measures the peak deviation from the function's central axis. It represents half the difference between the maximum and minimum values of the function.

To find the amplitude, use the formula:

\[\text{Amplitude} = \frac{\text{Maximum value} - \text{Minimum value}}{2} \]

For example, in Part a) of the exercise, the maximum value is 24, and the minimum value is 6. Plug these values into the formula:

\[\text{Amplitude} = \frac{24 - 6}{2} = 9\]

Similarly, in Part b), with a cosine function, the maximum value is 4 and the minimum value is -16:

\[ \text{Amplitude} = \frac{4 - (-16)}{2} = 10 \]

Finally, in Part c), where an electron oscillates, the maximum value is 10 and the minimum value is -10:

\[ \text{Amplitude} = \frac{10 - (-10)}{2} = 10 \]

Understanding this concept is key to solving sine and cosine function problems. The amplitude is always a positive number and indicates the height of the wave from its center.

Determining Sinusoidal Axis

The sinusoidal axis is the central horizontal line around which the function oscillates. It represents the average of the maximum and minimum values of the function.

To determine the sinusoidal axis, use the formula:

\[ \text{Sinusoidal Axis} = \frac{\text{Maximum value} + \text{Minimum value}}{2} \]

Let's take Part a) of the exercise. With a maximum value of 24 and a minimum value of 6, the calculation is:

\[\text{Sinusoidal Axis} = \frac{24 + 6}{2} = 15 \]

For Part b), where the maximum value is 4, and the minimum value is -16, the calculation is:

\[ \text{Sinusoidal Axis} = \frac{4 + (-16)}{2} = -6 \]

In Part c), involving an electron's oscillation, the maximum is 10, and the minimum is -10. The sinusoidal axis is:

\[ \text{Sinusoidal Axis} = \frac{10 + (-10)}{2} = 0 \]

This horizontal midpoint helps significantly in graphing sinusoidal functions and understanding their motion. Once you know the sinusoidal axis, you can also easily determine the vertical translation of the sine or cosine wave.

Finding Period of Function

The period of a sinusoidal function is the distance required for the function to complete one full cycle. It reveals how frequently the oscillations repeat. Understanding the period is crucial for graphing and analyzing these functions.

To find the period, the method varies based on the given details. In general, it's twice the interval between the first maximum and minimum points.

For Part a), the first maximum occurs at \(30^{\text{°}}\) and the first minimum at \(80^{\text{°}}\). The interval is:

\[ 80^{\text{°}} - 30^{\text{°}} = 50^{\text{°}} \]

The period is twice this interval:

\[\text{Period} = 2 \times 50^{\text{°}} = 100^{\text{°}} \]

In Part b), with a cosine function, the maximum is at 0 and the minimum at \(\frac{2\pi}{3}\):

\[ \text{Interval} = \frac{2\pi}{3} \]

The period is:

\[ \text{Period} = 2 \times \frac{2\pi}{3} = \frac{4\pi}{3} \]

For Part c), where the oscillation frequency is given as 50 times per second, the period is the reciprocal of the frequency:

\[\text{Period} = \frac{1}{50} \text{ seconds} \]

Remembering these methods will help you quickly and accurately determine the period for any sinusoidal function you encounter.