Particle Held In The Air By String

We know that the horizontal components must equal zero as the particle is at rest, we also know that the horizontal components must equal $10\text{ N}$ , again because the particle is at rest. Therefore we can use this information to make a triangle out of the vectors of force with $10\text{ N}$ .

From here we can use the sine rule:

$\dfrac{T_1}{\sin(30^\circ)}=\dfrac{T_2}{\sin(40^\circ)}=\dfrac{10}{\sin(110^\circ)}$

Therefore

$T_1 + T_2 = \dfrac{10\sin(30^\circ)}{\sin(110^\circ)} + \dfrac{10\sin(40^\circ)}{\sin(110^\circ)} = \boxed{12.16}$