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