00:01
In this question, we are asked to find the points on the parameter curve, where the tangent line is horizontal and vertical.
00:10
And i made a type i forgot pi here.
00:13
Recode that the tangent line is horizontal if dy or dx equals to 0.
00:27
And it's vertical if dy over dx is either plus or minus infinity.
00:40
Alright.
00:42
And from here, we see that we need to calculate dy or dx.
00:46
And for parameter curves, d .y over dx equals to d .y or dt, divide by dx or dt.
00:57
In our case, dy or dt equals to cos theta, minus, and by the product rule, the derivative of theta cause theta equals to cos theta minus theta multiplied by sine theta.
01:17
That's dy over dt.
01:18
Dx or dt equals to negative sine theta plus sine theta plus theta multiplied by cos theta.
01:34
Alright.
01:35
Now we can see that cos theta cancels in the numerator and sine theta cancelled in the denominator and we are going to get negative, we are going to get theta times sine theta in the numerator and we are going to get theta times cos theta in the denominator.
02:05
Next we can cancel theta and we are going to get tangent theta.
02:14
That's d y over d x.
02:18
Now for the tangent line to be horizontal, we want it to be equal to 0.
02:26
We want d -y or dx to be 0.
02:31
And this means we want tangent theta to be equal to 0.
02:42
And tangent theta equals to 0 where sine equals to 0 and sine equals to 0 when theta equals to negative.
02:51
So, note that in the interval, negative 2 pi and 2 pi are included.
02:57
So, which means that we're going to get 0 at negative 2 pi because sine is 0, sign equals to 0 at negative 2 pi, at negative pi at 0, at pi and 2 pi.
03:14
So these are five points where sine equals to 0, and therefore tangent equals to 0.
03:21
And therefore, these are the values of theta, where the tangent line is horizontal.
03:26
Now we need to calculate the values of x and y.
03:29
So we can create a table.
03:37
So the first line is going to be theta, and in the second line is going to be x.
03:49
And the third line is going to be y.
03:52
And now i'm going to copy the formulas for x and y.
04:04
All right.
04:04
Now, when we plug in theta equals to negative 2 pi, sine is going to be 0, and cosine is going to be 1.
04:12
So we are going to get x equals 1...