Question

Designer functions Design a sine function with the given properties. It has a period of 24 hr with a minimum value of 10 at $t=3$ hr and a maximum value of 16 at $t=15\mathrm{hr}.$

Short answer

Answer: The equation of the designed sine function is $f(t)=3\sin (\frac{\pi }{12}(t-3))+13$.

Step-by-step solution

Find the amplitude

The amplitude is half the difference between the minimum and maximum values. In this case, the maximum value is 16 and the minimum value is 10. Thus, the amplitude (A) can be calculated as follows: $A=\frac{1}{2}(16-10)=3$

Find the vertical shift

The vertical shift is the average of the minimum and maximum values. In this case, the vertical shift (D) can be calculated as follows: $D=\frac{1}{2}(16+10)=13$

Find the angular frequency

The angular frequency (ω) is calculated using the period (P): $$\omega =\frac{2\pi }{P}$$ Given that the period is 24 hours, the angular frequency is: $$\omega =\frac{2\pi }{24}=\frac{\pi }{12}$$

Find the horizontal shift

The function reaches its minimum value at $t=3$, so we will use this to find the horizontal shift (C). We know that the sine function reaches its minimum value when the function is shifted by $\frac{3\pi }{2}$ radians. Using the information about the minimum value, we can construct an equation involving the angular frequency and the horizontal shift as follows: $$\frac{\pi }{12}(3-C)=\frac{3\pi }{2}$$ Solving for the horizontal shift (C), we get: $$C=3-\frac{12}{\pi }\times \frac{3\pi }{2}=-3$$

Write the sine function

Now that we have all the necessary parameters, we can write the sine function: $$f(t)=A\sin (\omega t+C)+D$$ Substitute the values of A, ω, C, and D: $$f(t)=3\sin (\frac{\pi }{12}(t-3))+13$$ So the designed sine function with the given properties is: $$f(t)=3\sin (\frac{\pi }{12}(t-3))+13$$

Key concepts

Sine Function

The sine function is one of the fundamental trigonometric functions and is essential in modeling periodic phenomena. It is expressed as $\sin (\theta )$, where $\theta$ represents an angle measured in radians. The sine function is periodic and oscillates between -1 and 1. This characteristic makes it incredibly useful for describing waves, cycles, and other repeating patterns.
When crafting a sine function, you can adjust various parameters to fit the specifics of the scenario you are modeling. These include:

• Amplitude, which defines the height of the waves.
• Period, which dictates how long one complete cycle of the function takes.
• Angular frequency, tying the function's speed to radians per unit of time.
• Horizontal and vertical shifts, positioning the wave along the axes.

The basic format of a sine function can be expressed as $f(t)=A\sin (\omega t+C)+D$, representing these parameters.

Amplitude

Amplitude refers to the height of the wave in a trigonometric function, specifically the sine function. It is calculated as half the difference between the maximum and minimum values of the function. In our exercise, the maximum value of the function is 16, and the minimum value is 10. To find the amplitude $A$, we use the formula: \ $$A=\frac{1}{2}(\text{max}-\text{min})$$ For this specific sine function, we find: \ $$A=\frac{1}{2}(16-10)=3$$
The amplitude of 3 means that the wave rises to 3 units above and sinks to 3 units below its average value (or vertical shift). A higher amplitude would result in taller peaks and deeper troughs, whereas a lower amplitude would make the wave flatter.

Period

The period of a sine function is the time it takes to complete one full cycle of its wave pattern. It measures how long it takes for the function to start repeating its values. In our specific case, the period is given as 24 hours, meaning a full cycle of the sine wave is completed within this timeframe.
To relate the period to the components of the sine function, we understand that the angular frequency is directly tied to the period. The formula to relate the period $P$ to angular frequency $\omega$ is:

• $\omega =\frac{2\pi }{P}$

For example, with a period of 24 hours, the angular frequency is calculated as $\omega =\frac{2\pi }{24}=\frac{\pi }{12}$. This conversion is crucial for determining the rate of oscillation in radians per unit time.

Angular Frequency

Angular frequency reflects how rapidly the sine function oscillates, often described in terms of radians per unit time. It directly correlates with the period of the function, providing a means of converting periodic time units into angular dimensions.

• Angular frequency $\omega$ is calculated using the formula: \ $$\omega =\frac{2\pi }{\text{Period}}$$

In the given exercise, the sine function's period is 24 hours, leading to an angular frequency $\omega$ as follows: \ $$\omega =\frac{2\pi }{24}=\frac{\pi }{12}$$
This implies that the function cycles through $\frac{\pi }{12}$ radians every hour. Angular frequency helps in adjusting the frequency of oscillations to align with real-world periodic behaviors, such as day-night cycles represented in hours.