00:02
So in this problem, we're given this diagram called the witch of maria agnese, and we're asked to show that the parametric equations for the curve, which is the position of this point p out here, okay, our x equals 2a cotangent theta, and y equals 2a, sum, squared theta okay so this is what we're trying to prove here all right so let us first notice on this diagram a few other relationships here i'm gonna call this point e up here where the line y equals 2a intersects the vertical axis okay and notice that this is a right angle here and since this is theta down here where this blue line through the point a to c from the origin is an angle theta with the horizontal axis, then this is also angle theta right there.
01:36
I'm going to call this point b right here where i went from the point a all the way down to the x axis.
01:44
And i'm going to call this point d, where i go from point b all the way down to the horizontal axis here.
01:56
Okay.
01:57
I can also notice that this angle, since this is a right triangle here, or a right angle here at the origin, this is pi over to minus theta, isn't it? okay.
02:15
Now, let's notice something.
02:23
Here next the distance from c to d well that's just a straight line all the way up from the x axis all the way up to this line y equals 2a so that means that distance is 2a and we can notice that the angle from c through the origin to d is theta we've labeled that one theta down there that was given to us so then we see that we have this big right triangle angle out here don't we and so we can write that the tangent of theta is what well tangent opposite over adjacent so that's the distance from c to d over the distance from zero to d okay which means that the cotangent of theta is the distance from zero to d over the distance from c to d then in it because cotangent is one over the tangent.
03:35
All right.
03:36
So this means we now have that the distance from zero to d is the distance from c to d times the cotangent of theta.
03:55
But the distance from c to d, we just said over here was 2a.
04:00
So this is 2a, co tangent theta.
04:05
And the distance from zero to d is the x coordinate and so that one does hold we just proved it and just showed it okay so next what can we see we can see that the distance from zero to a is what from zero to a okay that's this leg on the bottom of this right triangle, isn't it? and so that's the distance from o to e times cosine of this angle right here, isn't it? pi over 2 minus theta.
05:13
All right...