We know that a circuit has a alternating voltage of 100 volts that peaks every 0.5 second. We want to find a sinusoidal model for the voltage V as a function of the time t. To do this, we will consider the general forms of transformed .
Sine Function:& y= asin b(x- h)+ k
Cosine Function:& y= acos b(x- h)+ k
In these functions | a| is the , 2Ď€ b is the ( b>0), h is the horizontal , and k is the vertical shift. To write a sine or cosine function that models a , we will follow four steps.
- Identify any vertical shift k from the graph.
- Determine whether the graph should be modeled by a sine or a cosine function, and identify any horizontal shift h.
- Find the period and the value of b.
- Find the and use them to find the amplitude and the value of a.
Let's follow these steps one at a time!
Step 1
Let's look at the given graph.
We can see that the graph is centered at the x-axis and therefore, it is a sinusoid curve with no vertical shift, k = 0.
Step 2
Notice that the given graph crosses the y-axis at y=0.
Because the function passes through the , the graph is a sine curve with no horizontal shift. This means that h= 0.
Step 3
The shortest repeating portion of a sinusoid is called cycle. The horizontal length of each cycle is called period. Let's find the period of the given graph.
We can see that the graph repeats every 12 units. This means that the period is 12. The period of the graph of sine functions is 2Ď€ b. With this information, we can write an equation in terms of b.
1/2 = 2Ď€/b
Let's solve this equation for b!
1/2 = 2Ď€/b
1/2b= 2 π
b= 4 π
Step 4
Next, to find the amplitude, we will again look at the given graph. We can see that the maximum value is 100 and the minimum value is - 100.
Amplitude is half the difference of the maximum and minimum values.
| a|= maximum value-minimum value/2
In our case, the maximum value is 100 and minimum value is - 100.
| a|= 100-(- 100)/2
We will evaluate this formula to find the amplitude.
|a|= 100-(- 100)/2
|a| = 100
Equation of the Sinusoidal Model
Because of the location of the y-intercept, we concluded that the voltage should be modeled by a sine function. We also found that a= 100, b= 4Ď€, h= 0, and k= 0. With this information, we can finally write the equation of the model.
V= 100sin 4Ď€(t- 0)+ 0
⇕
V=100sin 4Ď€ t
