calculate values of t in an interval for which the value x(t) is zero

[iheartmaths]

calculate values of t in an interval for which the value x(t) is zero

The function is x(t)=e^-ktcos(2pift)
In this case, f= 3 Hz and k=0.7 m s^-1
Putting f in gives cos(6pit)=0
The interval is between t=0 and t=1.

From the graph given, x(t)=0 at six points in this interval. (graph plotting software gives 1/12, 3/12 up to 11/12)


From what I know:
the exponential can only be 0 at infinity so it can be ignored.
The period is 2pi/|b|; or 2pi/6pi in this case, giving 1/3.
I also know cos(x)=0 when x=npi/2, where n is an odd integer.

At this point it has thrown me a bit, just having an issue marrying it all together. I was tempted to arccos both sides which removes cos from the LHS, giving 6pit=1/2pi and then dividing both sides by 6pi to give 1/12 but that only gives one result and I get the feeling that this calculation is a bit misleading.

I've got a feeling I'm missing something obvious an will probably kick myself lol. A push in the right direction will be very much appreciated!

Thanks in advance

 

[JeffM]

calculate values of t in an interval for which the value x(t) is zero
The function is x(t)=e^-ktcos(2pift)
In this case, f= 3 Hz and k=0.7 m s^-1
Putting f in gives cos(6pit)=0
The interval is between t=0 and t=1.

From the graph given, x(t)=0 at six points in this interval. (graph plotting software gives 1/12, 3/12 up to 11/12)


From what I know:
the exponential can only be 0 at infinity so it can be ignored.
The period is 2pi/|b|; or 2pi/6pi in this case, giving 1/3.
I also know cos(x)=0 when x=npi/2, where n is an odd integer.

At this point it has thrown me a bit, just having an issue marrying it all together. I was tempted to arccos both sides which removes cos from the LHS, giving 6pit=1/2pi and then dividing both sides by 6pi to give 1/12 but that only gives one result and I get the feeling that this calculation is a bit misleading.

I've got a feeling I'm missing something obvious an will probably kick myself lol. A push in the right direction will be very much appreciated!

Thanks in advance

Why are you worried about t = infinity when your interval is (0, 1).

In any case, provided the exponential is not infinity and the cosine is zero, their product will be zero.

What is your question again?

 

[iheartmaths]

Why are you worried about t = infinity when your interval is (0, 1).

In any case, provided the exponential is not infinity and the cosine is zero, their product will be zero.

What is your question again?

I wasn't worried that t=infinity? I know ab=0 if a=0 or b=0, which is why I said the exponential could be ignored in the OP.

The question is the thread title: Calculate values of t in an interval for which the value x(t) is zero.

I guess I've solved cos(6pit) for t to get one result but I need to find all values between the interval, which is the bit I'm stuck on. I'm guessing it has something to do with the period but I've honestly drawn a blank.

 

[JeffM]

[MATH]k \text { is an integer, and } t = \dfrac{(2k - 1)}{12} \implies cos( 6 \pi t) = WHAT?[/MATH]
[MATH]k \le 0 \implies t \le WHAT?[/MATH]
[MATH]k \ge 6 \implies t \ge WHAT?[/MATH]