Question

Use a 30degrees and -60degrees right triangle to find the exact value of the following trigonometric expression. Cot 30 degrees=

Answer 1

The dirty little secret about trig is students mostly just need to understand the 45/45/90 right isosceles triangle and the 30/60/90 triangle, which is half an isosceles triangle.  

I'm sure you have seen it many times already, so I'll remind you in the 30/60/90 right triangle the sides opposite those angles are respectively in the ratio $1: \sqrt{3}:2$.  Remember the smallest side is always opposite the smallest angle, etc.   These make a right triangle because

$1^2 + \sqrt{3}^2 = 2^2$

So in this hypotenuse 2 triangle, the length 1 side is opposite the 30 degree angle and the $\sqrt{3}$ side is adjacent to it.

We could jump right to the answer but let's enumerate all the trig functions.  Really you should memorize the sine and cosine so you can get these quickly.

$\cos 30^\circ = \dfrac{ \textrm{adjacent} }{ \textrm{hypotenuse} } = \dfrac{\sqrt{3}}{2}$

$\sin 30^\circ = \dfrac{ \textrm{opposite} }{ \textrm{hypotenuse} } = \dfrac{1}{2}$

$\tan 30^\circ = \dfrac{ \textrm{opposite} }{ \textrm{adjacent} }= \dfrac{1}{\sqrt{3}} =\dfrac{\sqrt{3}}{3}$

$\cot 30^\circ = \dfrac{ \textrm{adjacent} }{ \textrm{opposite}} = \dfrac{\sqrt{3}}{1} =\sqrt{3}$

$\sec 30^\circ = \dfrac{ \textrm{hypotenuse} }{ \textrm{adjacent}}= \dfrac{2}{\sqrt{3}}$

$\csc 30^\circ = \dfrac{ \textrm{hypotenuse} }{ \textrm{opposite}}= \dfrac{2}{1} = 2$

We want the cotangent, another way to get it is

$\cot 30^\circ = \dfrac{\cos 30^\circ}{\sin 30^\circ} = \dfrac{\sqrt{3}/2}{1/2} = \sqrt{3}$

Answer 2

Remark
Tan 30 degrees  = opposite divided by the  adjacent
Sin 30 degrees = opposite divided by the hypotenuse =  1/2
You need the adjacent side. Use the Pythagorean theorem to find it.

Adjacent side
a^2 + b^2 = c^2
a = 1
c = 2
1^2 + b^2 = 2^2
1 + b^2 = 4  Subtract 1 from both sides.
b^2 = 4 - 1
c^2 = sqrt(3)

Find the Tan(30) 
The Tan(30) = opposite divided by the adjacent = 1 / sqrt(3) Normally, you would rationalize the denominator, but in this case I wouldn't bother.

Find the cot(30)
Cot(30) = 1/ Tan(30) 
Cot(30) = 1/(1/sqrt(3)) = sqrt(3)