00:02
We are going to find an equation of the tangent line to the curve at the point corresponding to the given value of the parameter t.
00:11
The curve is described as x equal sine of 7 t plus cosine of t and y equal cosine of 7t minus sine of t and the value of the parameter t is pi.
00:28
So we want to calculate the derivative of y respect to x and for that we do the following and first gotta say that we want to find that because knowing the derivative of a curve at a point then we know the slope of the turning line to the curve at that point so that's why we need derivative of y respect to x and now if we apply the chain rule because y and x both depends on the parameter of t we get that this is equal to the derivative of y respect to t times derivative of t respect to x which becomes formerly equal to derivative of y respect to t over derivative of x respect to t so if we calculate the derivative of y and x respect to t and do this caution here we get the derivative of y respect to x so we're going to do that derivative of y here respect to t is negative 7 sine of 7 times t minus cosine of t.
01:52
Now the derivative of x here we're respect to t is the derivative of sign of 7t respect to t is 7 negative 7 cosine of 7 t.
02:03
Here we have applied the same rule as we did up here.
02:11
Sorry it's plus it's positive.
02:14
It's derivative of sign is is cosine, so here it's positive.
02:19
Cosine of 70 and times the derivative of 7t, respective to 7, so this is correct.
02:25
And here, the next term, derivative of cosine of t is negative sine of t.
02:32
So this is the expression for the derivative of y respect to x, which is represented in terms of parameter of g.
02:44
So this derivative corresponding to the parameter t equal pi is negative 7, sine of 7 times pi minus cosine of pi over 7, cosine of 5, over 7, cosine of 7 pi minus sign of pi.
03:12
So, sign of any multiple of pi is 0, so here this is 0, and here up this is 0.
03:21
And then we have cosine of pi is negative 1 so in the numerator we get 1 because we have a negative sign here and cosine of 7 pi is negative 1 so we get negative 7 so the derivative of y respect to x for the value of the parameter t equal pi is negative 1 over 7 and this is the slope of the turning line to the curve at the given point tangent line now we get to calculate, which is the point where we are going to calculate that tangent line.
04:09
The corresponding point of tangency is, let's say, x, y equal for t equal pi.
04:28
So we get to apply these formulas here for x and y.
04:32
Sorry, by putting t equal pi.
04:35
So we get sine of 7 pi minus cosine of pi.
04:48
That is x and then y is cosine of 7 pi minus sign of pi and that is you know this is 0 this is 0 cosine of pi negative 1 so we get here we get here 1 2 we get 1 sorry made a mistake here i put the incorrect sign sorry let's verify that what i'm saying here we have a plus an x...