Question

A 200 N mass is supported by two wires that make angles of 54 degrees and 67 degrees with the ceiling.
Determine the tension in each wire.

Answer 1

Make 2 diagrams, one called the position diagram, which shows the
actual situation, marking the angles at the ceiling and showing the hanging
mass of 200 N.
the other is a vector diagram where the sides of the triangle show the tension
I labelled mine ABC where angle A = 23°, angle B = 121° and angle C = 36°
with AC = 200
by the sine law:
AB/sin36 = 200/sin121
AB = 200sin36/sin121 = 137.146 N

BC/sin23 = 200/sin121
BC= 200sin23/sin121 = 91.168 N

check my arithmetic

Answer 2

All angles are measured CCW from +x-axis.
T1*sin36+T2*sin157 = 200
Eq1: 0.59T1 + 0.39T2 = 200.

T1*cos36 = -(T2*cos157)
0.81T1 = -(-.92T2)
T1 = 1.14T2.
In Eq1, replace T1 with 1.14T2 and solve for T2:
0.59*1.14T2 + 0.39T2 = 200
0.67T2 + 0.39T2 = 200
1.06T2 = 200
T2 = 188.7 N.
T1 = 1.14*188.7 = 215 N.