How to find the analytical expression from the graph?

Question:

The figure shows the graph of current changes with time. The current has the function, I(t) = A sin (Bt + C), (A > 0, B > 0, |C| < pi/2). The graph passes the point (0, 3/2). Find the function of current.

Solution:

To find the function of current, we need to determine these three elements: A - amplitude, B - frequency and C - initial phase. There is a relation between the minimum period T and the frequency B.
their relation is: T = 2pi/B. From the given sine graph, we can find the half period of the sine graph is the distance from -pi/18 to 5pi/18.

T/2 = 5pi/18 - (-pi/18)

= 5pi/18 + pi/18

= 6pi/18

= pi/3

so, T = 2pi/3

from T = 2pi/B

we get, B = 2pi/T

B = 2pi ÷ 2pi/3

= 2pi × 3/2pi

= 3

Thus, we get the function I(t) = A sin (3t + C), now we will determine the initial phase C. For a sine graph y = sin x, when x = 0, y = 0. That is, sin 0 = 0 which corresponding to the point where t = -pi/18.
That is, 3 × (-pi/18) + C = 0 and I(-pi/18) = 0.

3 × (-pi/18) + C = 0

C = -3 × (-pi/18)

= pi/6

The initial phase of the graph is pi/6.

We get the function I(t) = A sin (3t + pi/6). Now we will determine the amplitude A. We are given that the graph passes the point (0, 3/2). That is, when t = 0, the value of the current, I(0), is 3/2. Substitute the
value of the point, I(0) = 3/2, into the function I(t) = A sin (3t + pi/6).

3/2 = A sin(3 × 0 + pi/6)

3/2 = A sin(pi/6)

3/2 = A × 1/2

so, A = 3

Thus, we get the function of the current I(t) = 3 sin (3t + pi/6). Watch the video for more details.