Given x(t)= e^(-kt) cos (2 pi f t), k=0.5s, f= 2hz, find when x(t)=0 for 0<=t<=1.

[Ryder100]

Hi there, I was hoping someone could help with this question. I will enter the information below:

x(t)= e^(-kt) cos (2 pi f t)

k=0.5s
f= 2hz

So I am being asked to calculate the values of t between t=0 and t=1 when the amplitude x(t) is 0.

It gives me a hint which says: consider values at which a cosine function is zero and also consider whether the exponential function can be zero.

 

[stapel]

x(t)= e^(-kt) cos (2 pi f t)

k=0.5s
f= 2hz

So I am being asked to calculate the values of t between t=0 and t=1 when the amplitude x(t) is 0.

It gives me a hint which says: consider values at which a cosine function is zero and also consider whether the exponential function can be zero.

Use what you learned back in algebra. When is a product equal to zero? When one or another of its factors is equal to zero. So, as the hint suggests, the product for x(t) will be zero whenever \(\displaystyle e^{-0.5t}\) is zero, and whenever \(\displaystyle \cos(4\pi t)\) is zero.

For the exponential, again, use what you learned back in algebra. When is the exponential ever zero?

For the cosine, use what you've learned in trig. When, on the first period, is the cosine equal to zero? So what must the argument, \(\displaystyle 4\pi t,\) be equal to? So what must t be equal to? To find all solutions, consider when, in full generality, is the cosine equal to zero? And so forth.