2021 Fall AMC 12B Problem 10

What is the sum of all possible values of t between 0 and 360 such that the triangle in the coordinate plane whose vertices are (\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ) is isosceles?

\textbf{(A)} \: 100 \qquad\textbf{(B)} \: 150 \qquad\textbf{(C)} \: 330 \qquad\textbf{(D)} \: 360 \qquad\textbf{(E)} \: 380

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.