Answer
(a) $T_A = 0.732~w$
$T_B = 0.897~w$
$T_C = w$
(b) $T_A = 2.74~w$
$T_B = 3.35~w$
$T_C = w$
Work Step by Step
(a) $T_C = w$
horizontal components:
$T_A~cos(30^{\circ}) = T_B~cos(45^{\circ})$
$T_A = \frac{T_B~cos(45^{\circ})}{cos(30^{\circ})}$
We can replace $T_A$ in the vertical equation.
vertical components:
$T_A~sin(30^{\circ}) + T_B~sin(45^{\circ}) = w$
$\frac{T_B~cos(45^{\circ})}{cos(30^{\circ})}~sin(30^{\circ}) + T_B~sin(45^{\circ}) = w$
$T_B~(cos(45^{\circ})~tan(30^{\circ})+sin(45^{\circ})) = w$
$T_B = \frac{w}{cos(45^{\circ})~tan(30^{\circ})+sin(45^{\circ})}$
$T_B = 0.897~w$
$T_A = \frac{(0.897~w)~cos(45^{\circ})}{cos(30^{\circ})}$
$T_A = 0.732~w$
(b) $T_C = w$
horizontal components:
$T_A~sin(60^{\circ}) = T_B~cos(45^{\circ})$
$T_A = \frac{T_B~cos(45^{\circ})}{sin(60^{\circ})}$
We can replace $T_A$ in the vertical equation.
vertical components:
$T_B~sin(45^{\circ})- T_A~cos(60^{\circ}) = w$
$T_B~sin(45^{\circ}) - \frac{T_B~cos(45^{\circ})}{sin(60^{\circ})}~cos(60^{\circ})= w$
$T_B~(sin(45^{\circ}) - cos(45^{\circ})~cot(60^{\circ})) = w$
$T_B = \frac{w}{sin(45^{\circ}) - cos(45^{\circ})~cot(60^{\circ})}$
$T_B = 3.35~w$
$T_A = \frac{(3.35~w)~cos(45^{\circ})}{sin(60^{\circ})}$
$T_A = 2.74~w$
