The curve x = 2 cos(t), y = sin(t) − sin(2t), 0 ≤ t ≤ 2π crosses itself once, find this intersection point and then find the equations of the two tangent lines at the intersection point.

You're looking for such that .



Recall that , so the second equation can be written as




Since , and assuming , you get




which admits two possibilities; either or . In the first case, since we're assuming , we can use the fact that to arrive at a solution of .

In the second case, you have



Let's check which of these solutions work. If , then the sine equation suggests . However,




so in fact this is an extraneous solution. So let's return to the first equation in the system,



Again, assuming , we can use the fact that to arrive at a solution of . Now, if , we get . Let's check if this works:




Indeed, this solution works! So the curve intersects itself at the point (1,0), which the curve passes for the first time through when and the second time when .

Now, to find the tangent line, we need to compute the derivative of with respect to . You have



When , you have a slope of ; at , the slope is .

The tangent lines are then