TL;DR: There is something wrong with the problem definition. The minimum force on the $T_{AC}$ is approximately 906 lbs at $\theta=65\deg$ (in that case $T_{AB}= 422[lbs]$)
NOTE: From (Eq.3) what you really get is that:
$$ \frac{T_{AC}}{ T_{AB}} =\frac{ cos (25)} {cos (θ)} $$
which actually can be interpreted, as:
- $T_{AC}>T_{AB}$ when $\theta > 25\deg$
- $T_{AC}<T_{AB}$ when $\theta < 25\deg$
So basically you need to check first what happens at 25$\deg$.
at 25$\deg$
At 25$\deg$, there is symmetry so:
$$T_{AC} =T_{AB}$$
So from the second equation you can obtain that:
$$ ∑ Fy = T_{AB}= T_{AC}= \frac{1000}{sin (25)}=2366$$
which means that the force is already exceeding 1000.
So you would need to have an angle $\theta >25$.
Because then $T_{AB}<T_{AC}$, and if $T_{AC}= 700 [lbs]$ then $T_{AB}$ will also be $T_{AB}<700$.
Solving for $\theta$.
Since $\theta >25\deg$, $T_{AC}>T_{AB}$ and therefore you can substitute in Eq.2, the following form of eq.3 $ T_{AB} =T_{AC}\frac{cos (θ)}{ cos (25)} $ :
$$T_{AB} sin (25) + T_{AC} sin (θ) = 1000$$
$$T_{AC}\frac{\cos (θ)}{ \cos (25)} sin (25) + T_{AC} \sin (θ) = 1000$$
$$T_{AC}\left(\frac{\cos (θ)}{ \cos (25)} sin (25) + \sin (θ)\right) = 1000$$
$$T_{AC}\left(\cos (θ)\tan(25) + \sin (θ)\right) = 1000$$
$$T_{AC}\left(\cos (θ)\tan(25) + \sin (θ)\right) = 1000$$
From this equation we can solve for $T_{AC} = 1000\frac{1}{\cos (θ)\tan(25) + \sin (θ)}$, and we can plot this equation.
The problem is that the minimum value of $T_{AC}$ is at approx 900[lbs].
The reason is that the maximum value for the expression $\cos (θ)\tan(25) + \sin (θ)$ is for $\theta= 65\deg$. So for 65 deg, you can get the minimum $T_{AC}\approx 906$. (In that case, $T_{AB} = 422 [lbs]$).
So, you can't really protect the AC rope from breaking.
if you only needed $T_{AB}$ to be under 700[lbs], then you can achieve that for angles $\theta \ge 45\deg$. see graph below
#%%
import numpy as np
import matplotlib.pyplot as plt
# %%
xs = np.linspace(25,85,100)
ts = xs*np.pi/180
Ts = 1000/(np.cos(ts)*np.tan(25*np.pi/180)+ np.sin(ts))
TABs = Ts*np.cos(ts)/np.cos(25*np.pi/180)
# %%
plt.figure(figsize=(10,8))
plt.plot(xs,Ts, label='$T_{AC}$')
plt.plot(xs,TABs, label='$T_{AB}$')
plt.grid()
plt.legend(fontsize=15)
plt.xlabel('Angle[deg]', fontsize=15)
plt.ylabel('Forces [lbs]', fontsize=15)
