00:04
Let's determine where the plane, y plus x equals 0, intersects the curve given with vector equation r of t equals cosine t, sine t, t, for values of t between and including 0 to 4 pi.
00:21
Well, to find out where, what values of t give us these points, we want to know where the y component and the x component, when added together, give us zero.
00:34
And so we are going to set up the equation, sine t plus cosine t equals zero.
00:54
That's going to be where sine t is equal to negative cosine t.
00:59
And if we think about the unit circle, and i don't try to draw very careful or precise unit circles just for reference, the places where the sine and the cosine are opposite values are at these points along the unit circle.
01:16
Negative root 2 over 2, positive root 2 over 2, m positive root 2 over 2, negative root 2 over 2.
01:27
So this happens first time around at 3 pi over 4, and then 7 pi over 4.
01:37
But we want to go two trips around the unit circle.
01:40
Here's going to go up to 4 pi.
01:43
And so these will repeat at 11 pi over 4.
01:49
And at 15 pi over 4.
01:55
And so the values of t that will give us the points where the plane intersects that curve are going to be t equals 3 pi over 4 t equals 7 pi over 4 or t equals 7 pi over 4 or t equals 11 pi over 4 or t equals 15 pi over 4...