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