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 period, frequency, time (phase) shift, and time constant of the given expression -3e^(-2*t)*sin(pi*t pi/3), we need to analyze the equation. Let's break it down step by step:
a) Period:
The period represents the length of one complete cycle of the function. For a sine function, the period is given by 2π divided by the coefficient of t in the function.
In this case, the coefficient of t is pi/3. Therefore, the period can be calculated as:
Period = 2π / (pi/3) = 6.
So, the period of the given function is 6.
b) Frequency:
The frequency is the reciprocal of the period and represents the number of cycles per unit of time. In this case, the frequency can be calculated as:
Frequency = 1 / Period = 1 / 6.
Therefore, the frequency of the given function is 1/6.
c) Time (Phase) Shift:
To determine the time (phase) shift, we need to identify any changes to the argument of the sine function.
In this case, the function is -3e^(-2*t)*sin(pi*t pi/3). The argument of the sine function inside the parentheses is pi*t pi/3.
The time (phase) shift represents the horizontal translation of the waveform. To determine it, we need to set the argument equal to zero and solve for t:
pi*t pi/3 = 0
Solving this equation, we get t = 0.
Therefore, the time (phase) shift of the given function is 0.
d) Time Constant:
The time constant represents the measure of time it takes for the function to decay or diminish. It is determined by the coefficient of t in the exponential term.
In this case, the coefficient of t is -2. The time constant can be calculated as:
Time Constant = 1 / (absolute value of coefficient of t) = 1 / |-2| = 1/2.
So, the time constant of the given function is 1/2.
To summarize:
a) The period is 6.
b) The frequency is 1/6.
c) The time (phase) shift is 0.
d) The time constant is 1/2.