00:02
So in this problem we give them this diagram called the witch of maria and agnes e.
00:09
And were asked to show that the parametric parametric equations for the curve which is the position of this point p.
00:19
Out here.
00:22
Okay.
00:23
Our x equals to a co tangent data and y equals to a sine squared data.
00:41
Okay? so this is what we're trying to prove here.
00:46
All right, so let us first notice on this diagram, a few other relationships here.
00:58
I'm gonna call this point e up here where the line y equals to a intersects the vertical axis.
01:09
Okay, i noticed that this is a right angle here.
01:15
And since this is data down here where this blue line through the point a.
01:23
To see from the origin is an angle theta with the horizontal axis.
01:30
Then this is also angle theta right there.
01:36
I'm gonna call this point b.
01:38
Right here where i went from the point a.
01:41
All the way down to the x axis.
01:45
And i'm going to call this point d.
01:49
Where to go from point b.
01:51
All the way down to the horizontal axis here.
01:56
Okay, i can also notice that this angle pence.
02:03
This is a right triangle here.
02:06
Or right angle here at the origin.
02:08
This is pi over 2- data, isn't it.
02:14
Okay, now let's notice something here.
02:24
Next the distance from c to d.
02:29
Well that's just a straight line all the way up from the x axis.
02:33
All the way up to this line y equals to a.
02:36
That means that distance is to a.
02:40
And we can notice that the angle from see through the origin to d.
02:49
His data we've enabled that one data down there that was given to us.
02:54
So then we see that we have this big right triangle out here don't we? and so we can write that the tangent of data is what well tangent opposite over adjacent.
03:11
So that's the distance from c.
03:13
To d.
03:14
Over the distance from 0 to d.
03:19
Okay.
03:20
Which means that the co tangent of theta is the distance from zero to d.
03:28
Over the distance from c.
03:29
To d...