the net effect:
R = 240 c o s θ + T c o s 30 ° + 400 c o s 60 ° = 600 R=240cos\theta+Tcos30\degree+400cos60\degree=600 R = 240 cos θ + T cos 30° + 400 cos 60° = 600
0 = 400 c o s 30 ° + T c o s 60 ° − 240 s i n θ − 360 0=400cos30\degree+Tcos60\degree-240sin\theta-360 0 = 400 cos 30° + T cos 60° − 240 s in θ − 360
240 c o s θ + 3 T / 2 = 400 240cos\theta+\sqrt{3}T/2=400 240 cos θ + 3 T /2 = 400
T / 2 − 240 s i n θ = 360 − 200 3 T/2-240sin\theta=360-200\sqrt{3} T /2 − 240 s in θ = 360 − 200 3
T / 2 = ( 400 − 240 c o s θ ) / 3 T/2=(400-240cos\theta)/\sqrt{3} T /2 = ( 400 − 240 cos θ ) / 3
( 400 − 240 c o s θ ) / 3 − 240 s i n θ = 360 − 200 3 (400-240cos\theta)/\sqrt{3}-240sin\theta=360-200\sqrt{3} ( 400 − 240 cos θ ) / 3 − 240 s in θ = 360 − 200 3
400 − 240 c o s θ − 240 3 s i n θ = 360 3 − 600 400-240cos\theta-240\sqrt3sin\theta=360\sqrt3-600 400 − 240 cos θ − 240 3 s in θ = 360 3 − 600
6 c o s θ + 6 3 s i n θ = 25 − 9 3 6cos\theta+6\sqrt3sin\theta=25-9\sqrt3 6 cos θ + 6 3 s in θ = 25 − 9 3
36 ( 1 − s i n 2 θ ) = 625 − 450 3 + 243 − 12 3 s i n θ ( 25 − 9 3 ) + 108 s i n 2 θ 36(1-sin^2\theta)=625-450\sqrt3+243-12\sqrt3sin\theta(25-9\sqrt3)+108sin^2\theta 36 ( 1 − s i n 2 θ ) = 625 − 450 3 + 243 − 12 3 s in θ ( 25 − 9 3 ) + 108 s i n 2 θ
144 s i n 2 θ − 12 3 s i n θ ( 25 − 9 3 ) + 832 − 450 3 = 0 144sin^2\theta-12\sqrt3sin\theta(25-9\sqrt3)+832-450\sqrt3=0 144 s i n 2 θ − 12 3 s in θ ( 25 − 9 3 ) + 832 − 450 3 = 0
s i n θ = 195.6 − 38265 − 4 ⋅ 144 ( 832 − 450 3 ) 288 = 195.6 − 38265 − 30284 288 = 195.6 − 89.3 288 = 0.3691 sin\theta=\frac{195.6-\sqrt{38265-4\cdot144(832-450\sqrt3)}}{288}=\frac{195.6-\sqrt{38265-30284}}{288}=\frac{195.6- 89.3}{288}=0.3691 s in θ = 288 195.6 − 38265 − 4 ⋅ 144 ( 832 − 450 3 ) = 288 195.6 − 38265 − 30284 = 288 195.6 − 89.3 = 0.3691
θ = 21.66 ° \theta=21.66\degree θ = 21.66°
T = 2 ( 400 − 240 c o s 21.66 ° ) / 3 = 204 T=2(400-240cos21.66\degree)/\sqrt3=204 T = 2 ( 400 − 240 cos 21.66° ) / 3 = 204
#333702 on Dec 2022
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