Align the 20-N force along the y-axis, then, to keep the body in equilibrium, the horizontal and vertical components of the two other forces must be equal:
20−12cosθ1−15cosθ2=0, 12sinθ1−15sinθ2=0. 122(1−cos2θ1)=152(1−cos2θ2), (15cosθ2)2−(12cosθ1)2−81=0. So, we have the upper and lower equations that form quite a pretty system of equations. Solve the system (you may use substitution 12cosθ1=x, 15cosθ2=y):
12cosθ1=7.975,15cosθ2=12.025,θ1=48.3°,θ2=36.7°.
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