Given the following question:
-3e^(-2*t)*sin(pi*t + pi/3)]
How do I find
a) period
b) frequency
c) time (phase) shift
d) time constant
To find the answers to the questions for the given function -3e^(-2t)*sin(pi*t + pi/3), we need to understand the properties of the function and use them to calculate the period, frequency, time shift, and time constant.
a) Period:
The period of a function is the length of one complete cycle. For a sine function in the form sin(bt), where b is a coefficient, the period is given by T = 2π/b.
In the given function, we have sin(pi*t + pi/3), so the coefficient is b = pi. Therefore, the period is T = 2π/pi.
b) Frequency:
The frequency of a function is the number of cycles it completes in one unit of time. It is the reciprocal of the period, f = 1/T.
So in this case, the frequency f = 1/(2π/pi) = pi/2.
c) Time (Phase) Shift:
The time shift, also known as the phase shift or horizontal shift, indicates how the function is shifted along the x-axis. It represents the amount by which the function has been horizontally translated. For a sine function of the form sin(bt + c), the time shift is given by -c/b.
In the given function, we have sin(pi*t + pi/3). Therefore, the time shift is - (pi/3) / pi = -1/3.
d) Time Constant:
The time constant represents the time it takes for the function to decay to 1/e of its initial value. In this case, the function has an exponential term -3e^(-2t), so the decay factor is e^(-2t).
To calculate the time constant, we need to find the value of t at which the exponential term equals 1/e. So we set e^(-2t) = 1/e and solve for t.
e^(-2t) = 1/e
Taking the natural logarithm on both sides:
-2t = -1
t = 1/2
Therefore, the time constant is 1/2.
To summarize:
a) Period = 2π/pi
b) Frequency = pi/2
c) Time (Phase) Shift = -1/3
d) Time Constant = 1/2