I think Melody made a slight error towards the end....I'm pretty sure the answer isn't EXACTLY 45, either.
(T*(cos(45)/cos(30)))*sin(30) + (T*sin(45) - 50 = 0 ???
When I stick "45" in for "T," we get
$$\left({\mathtt{45}}{\mathtt{\,\times\,}}\left({\frac{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}{\left({\mathtt{45}}^\circ\right)}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}{\left({\mathtt{30}}^\circ\right)}}}\right)\right){\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{30}}^\circ\right)}{\mathtt{\,\small\textbf+\,}}\left({\mathtt{45}}{\mathtt{\,\times\,}}\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{45}}^\circ\right)}{\mathtt{\,-\,}}{\mathtt{50}}\right) = {\mathtt{0.190\: \!978\: \!224\: \!309\: \!896\: \!5}}$$
Let's see what we can do....taking Melody's work from this point......
50√(6) / (1 + √(3))
Let's multiply by the whole fraction by the conjugate of (1 + √(3)) on "top" and "bottom"
We have
50√(6) / (1 + √(3)) * (1 - √(3)) / (1 - √(3))
This gives us
50√(6) * (1 - √(3)) / (-2)
Sinplifying the top, we have
[50√(6)- 50√(18)] / (-2)
Let's multiply top and bottom by (-1) to get rid of that pesky (-2).....Nore that the things in the numerator just switch places when we do this...so we have
[50√(18)- 50√(6)] / (2)....and dividing everything by 2, we get
25 (√(18)- √(6)).....we could simplify the "18" under the square root, but we would still have two square roots, so let's just leave it alone.
So "T" = 25 (√(18)- √(6)) .......BTW this is close to 45 !!!!!
And plugging in for "T" we have
25*(√(18)-√(6))*(cos(45)/cos(30))*sin(30)+(25*(√(18)-√(6))*sin(45)-50)
If you evaluate this answer inyour calculator (or the calculator on this site), you'll get something that ≈ 0
(Sorry....they've made some changes to this site and I haven't figured out how to evaluate things, just yet )
Anyway.....I feel pretty confident that this is the correct answer