Melody

And those other kids say their homework is hard and time-consuming. 

You have mastered the art of time and skill allocation. 

Clever you.

\(\left(6x+\dfrac{1}{3x}\right)^6\\~\\ T_n=\begin{pmatrix} 6\\ n \end{pmatrix} (\frac{1}{3x})^n(6x)^{n-6}\\ \text{If you pull out the x terms you get } (\frac{1}{x})^n*x^{n-6} \\ \text{The constant term will have }x^0\qquad\\ \text{So solve for n}\\ etc\)

Please no other answerer finish this. 

Thanks Heureka,

Here is another approach.    

\(\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{7\pi}{12} + \sin \frac{9\pi}{12} + \sin \frac{11\pi}{12}\\ =\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{\pi}{12}\\ =2(\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} )\\ =2(\sin \frac{\pi}{12} + \sin \frac{\pi}{4} + \cos \frac{\pi}{12} )\\ =2(\sin \frac{\pi}{12} + \cos \frac{\pi}{12} )+2\sin \frac{\pi}{4}\\ =2\sqrt{(\sin \frac{\pi}{12} + \cos \frac{\pi}{12} )^2}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(\sin^2 \frac{\pi}{12} + \cos^2 \frac{\pi}{12} +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(1 +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(1 +\sin \frac{\pi}{6})}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(1 +\frac{1}{2})}+ \frac{2}{\sqrt{2}}\\ =2\sqrt{(\frac{3}{2})}+ \frac{2}{\sqrt{2}}\\ =2(\frac{\sqrt3}{\sqrt2})+ \frac{2}{\sqrt{2}}\\ =\sqrt6+\sqrt2\)

LaTex

\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{7\pi}{12} + \sin \frac{9\pi}{12} + \sin \frac{11\pi}{12}\\

=\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{5\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{\pi}{12}\\

=2(\sin \frac{\pi}{12} + \sin \frac{3\pi}{12} + \sin \frac{5\pi}{12} )\\

=2(\sin \frac{\pi}{12} + \sin \frac{\pi}{4} + \cos \frac{\pi}{12} )\\

=2(\sin \frac{\pi}{12}   + \cos \frac{\pi}{12} )+2\sin \frac{\pi}{4}\\

=2\sqrt{(\sin \frac{\pi}{12}   + \cos \frac{\pi}{12} )^2}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(\sin^2 \frac{\pi}{12}   + \cos^2 \frac{\pi}{12} +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(1 +2\sin \frac{\pi}{12}\cos \frac{\pi}{12} )}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(1 +\sin \frac{\pi}{6})}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(1 +\frac{1}{2})}+ \frac{2}{\sqrt{2}}\\

=2\sqrt{(\frac{3}{2})}+ \frac{2}{\sqrt{2}}\\

=2(\frac{\sqrt3}{\sqrt2})+ \frac{2}{\sqrt{2}}\\

=\sqrt6+\sqrt2

You do not need a calculator, plot them by hand.

You can plot them here I guess.  I have done the first one for you.

Yes, although I would go for 5:45 (I mean 45 minutes) myself

You are very welcome. 

I come here to help people learn and I think I have done that here so I am happy.   

Get t in terms of x, I mean make t the subject, then substitute it into the second equation so that the t disappears.