00:01
Consider the given parametric equations below, we have x of t, which is equal to 8 sine of t, and y of t, which is equal to 4 cosine of t.
00:09
For the first part, you want to find the equation for the line that's tangent to this curve at t which is equal to pi over 4.
00:20
And if we recall, equation of the tangent line to the curve at t is given by y that's equal to m sub t, times x minus x of 0 plus y sub 0 where m sub t is d y over d x evaluated at the given value of t x x of 0 is equal to x of t and our y sub 0 is y of t.
00:53
So first you want to get the values of x of 0 and y sub 0.
00:59
Now at t equal to pi over 4, we have x of pi over 4 equal to 8 sign of pi over 4, which is 8 times square root of 2 over 2, or 4 squared of 2.
01:20
And y of pi over 4 is 4 cosine of pi over 4.
01:27
That's 4 times squared of 2 equal to 2 squared of 2.
01:32
And for m sub t, we first want to get d .y over dx.
01:37
So since d .y over dx is d.
01:42
Y over d .t over dx over dt, that's y prime of t over x prime of t equal to 8 cosine of t over negative 4 sine of t, which is equal to negative 2, cotangent of t, then at t equals pi 4, m sub t, which is d y over d x, evaluated at t equals 5 or 4, will be equal to negative 2 cotangent of pi over 4.
02:23
That's negative 2 times 1 over tangent of pi over 4.
02:29
That's equal to negative 2 because tangent of 5 over 4 is 1...