# Urgent help neaded with function of sin

• Sep 18th 2010, 04:00 AM
CoyoteGrin
Urgent help neaded with function of sin
Okay, so here's what I've got:

Consider the function f(x)=sin(wx), where w is a positive constant.
If f(a) = 0 and f(b) = 1, what is the minimum possible distance between a and b?

I already have the answer to this problem (pi as in the number over 2w), however I have no idea what the process is to get to it.

If anybody knows how to solve this, please post a step-by-step guide on how to do so.
Thanks
• Sep 18th 2010, 04:17 AM
Quote:

Originally Posted by CoyoteGrin
Okay, so here's what I've got:

Consider the function f(x)=sin(wx), where w is a positive constant.
If f(a) = 0 and f(b) = 1, what is the minimum possible distance between a and b?

I already have the answer to this problem (pi as in the number over 2w), however I have no idea what the process is to get to it.

If anybody knows how to solve this, please post a step-by-step guide on how to do so.
Thanks

$Sin(0+2n{\pi})=0,\;\;Sin\left(\frac{{\pi}}{2}+2n{\ pi}\right)=1,\;\;Sin({\pi}+2n{\pi})=0$

$Sin(a)=0,\;\;Sin(b)=1\Rightarrow\ (b-a)_{min}=\frac{{\pi}}{2}$

$\Rightarrow\ w\left(x_2-x_1\right)_{min}=\frac{{\pi}}{2}$

Therefore, minimum distance in degrees is $x_2-x_1=\frac{{\pi}}{2w}$
• Sep 18th 2010, 07:31 AM
CoyoteGrin
Thank you for posting, Archie. Unfortunately, I'm about as much as a layman now in math as one can get, so I'm still a little confused.

1.)You posted Sin(0+2n(pi)) = 0 and Sin((pi/2)+2n(pi)) = 1. Where did the 0 and the pi/2 come from? Are these just something that you have to memorize or use a calculator for, or is there a way to figure out what to add to 2n(pi) to get any number in a sin equation?

2.) It's obvious that "min" means "minimum," but why on earth put it there?

3.) I noticed that your answer is (pi)/2, whereas mine is (pi)/(2w). How is it that, in your answer, the "w" is somehow irrelevant?

I apolagize for my ignorance (hopefully you got a chuckle out of it), but it's been years since I had trigonometry and even the more basic stuff is barely still in my memory.
Again, thank you
• Sep 18th 2010, 08:13 AM
sa-ri-ga-ma
If you draw a graph sin(ωx) vs ωx, at x = a, f(a) = sin(ωa) = 0. So ωa = 0

At x = b, f(b) = sin(ωb )= 1. So ωb = π/2.

So minimum distance ωb - ωa = π/2

And (b-a) = π/2ω
• Sep 18th 2010, 09:55 AM
Quote:

Originally Posted by CoyoteGrin
Thank you for posting, Archie. Unfortunately, I'm about as much as a layman now in math as one can get, so I'm still a little confused.

1.)You posted Sin(0+2n(pi)) = 0 and Sin((pi/2)+2n(pi)) = 1. Where did the 0 and the pi/2 come from? Are these just something that you have to memorize or use a calculator for, or is there a way to figure out what to add to 2n(pi) to get any number in a sin equation?

2.) It's obvious that "min" means "minimum," but why on earth put it there?

Because the function is periodic and f(b)-f(a)=1 for ever-increasing angles

3.) I noticed that your answer is (pi)/2, whereas mine is (pi)/(2w). How is it that, in your answer, the "w" is somehow irrelevant?

No, I left the expression as a product, from there we can just divide both sides by w.

I apolagize for my ignorance (hopefully you got a chuckle out of it), but it's been years since I had trigonometry and even the more basic stuff is barely still in my memory.
Again, thank you

Hi CoyoteGrin,

You need an awareness of the graph of the function, or the "unit circle", to make sense of the question.

Also, I had a typo in my first post!! sorry(Doh)

Starting at $x=0^o,$ Sinx rises smoothly from 0 to 1, when $x=90^o$ or $\frac{{\pi}}{2}$

Then the graph drops to zero when x reaches 180 degrees, continues to fall to -1 when x=270 degrees,
then rising back to zero when x reaches 360 degrees.

If however, the function is Sin(wx), then this will rise from 0 to 1 as wx goes from 0 degrees to 90 degrees.

90 degrees is the minimum distance (in degrees) required to go from 0 to 1.
It's the minimum because the graph repeatedly reaches 1 as we cycle through multiples of 360 degrees.

Hence $(b-a)_{min}=\frac{{\pi}}{2}$

$w\left(x_2-x_1\right)=\frac{{\pi}}{2}$

$x_2-x_1=min\ distance\ in\ degrees=\frac{{\pi}}{2w}$