00:01
In this question, we have parametric equations.
00:02
X of t equals the cosine of t, and y of t equals the sine of t times the cosine of t, for t values from 0 to 2 pi.
00:11
And this will describe the curve that's shown below.
00:14
In part a, we're going to find the points, which means the x and y values where the tangent lines to the curve are horizontal.
00:22
So what do we do? we know that dy dx is equal to the derivative of y with respect to t divided by the derivative of x with respect to t.
00:36
So if i am going to look for horizontal tangencies, so a horizontal tangency occurs when the slope of the tangent line is equal to 0.
00:53
So when dy dx is equal to 0.
00:57
Now, dy dx is a fraction this time.
01:00
The only way that a fraction can equal 0 is if the numerator is equal to 0.
01:04
So this implies that dy dt is equal to 0.
01:09
Now, my dy dt this time requires the product rule.
01:14
I'm going to have my first factor, sine of t, times the derivative of the second factor, negative sine of t, plus my second factor, plus my second factor, cosine of t, times the derivative of my first factor, which is cosine of t, equals 0.
01:35
And this is giving us negative sine squared t plus the cosine squared of t is equal to 0.
01:46
Now, you may remember that negative sine squared t plus cosine squared of t, that's actually an identity for precalculus.
01:56
It is the cosine of 2t.
01:59
So the cosine of 2t is equal to 0.
02:04
Now, this implies that 2t itself is of the form pi over 2 plus pi n, so that t itself looks like pi over 4 plus pi over 2n.
02:21
In the interval from 0 to 2 pi, that gives you pi over 4, 3 pi over 4, 5 pi over 4, and 7 pi over 4.
02:34
So i'm going to take each of these t values and figure out what the corresponding xy point is.
02:41
So i'm taking t equals pi over 4, t equals 3 pi over 4, t equals 5 pi over 4, and t equals 7 pi over 4.
02:56
And we're going to figure out what is the corresponding xy point.
03:08
So let's see.
03:09
If t equals pi over 4, my x is cosine of t.
03:16
Cosine of pi over 4 is square root 2 over 2.
03:21
Now, how about the y? that'll be root 2 over 2 times root 2 over 2.
03:29
That's 2 fourths or 1 half.
03:34
Now, when t is 3 pi over 4, x is the cosine of 3 pi over 4, which is negative square root 2 over 2.
03:44
While my y is going to be root 2 over 2 times negative root 2 over 2...