Question

How do you find the cot(4pi/3)? I have graphed it and labeled my hypotenuse but I am struggling how to figure out the length of the hypotenuse, etc.

Answer 1

length of the hypotenuse is always 1

Answer 2

find \(\frac{4\pi}{3}\) on the unit circle on the last page of the attached cheat sheet
the first coordinate is cosine, the second coordinate is sine

Answer 3

and \(\cot(\frac{4\pi}{3})=\frac{\cos(\frac{4\pi}{3})}{\sin(\frac{4\pi}{3})}\)

Answer 4

\[cot\left(\frac{4\pi}{3}\right)=\frac{1}{tan\left(\frac{4\pi}{3}\right)}=\frac{1}{tan\left(\pi+\frac{\pi}{3}\right)}=\frac{1}{tan\left(\frac{\pi}{3}\right)}=\frac{1}{\gamma}\]
YOu require a special triangle:

So:
\[\gamma=\tan\left(\frac{\pi}{3}\right)=\sqrt{3}\]
Therefore:
\[cot\left(\frac{4\pi}{3}\right)=\frac{1}{\gamma}=\frac{1}{\sqrt{3}}=\sqrt{\frac{1}{3}}\]