Answer
- $\frac{\sqrt 3}{3}$
Work Step by Step
To find exact value of $\cot 480^{\circ}$, let's find its co-terminal angle between $0^{\circ}$ and $360^{\circ}$ first-
Co-terminal angle of $480^{\circ}$ = $480^{\circ} - 360^{\circ}$ = $120^{\circ}$
Therefore
$\cot 480^{\circ}$ = $\cot 120^{\circ}$
($480^{\circ}$ and $120^{\circ}$ are co-terminal and trigonometric functions of co-terminal angles are same)
Now to calculate exact value of $\cot 120^{\circ}$, let's find its reference angle first. As $ 120^{\circ}$ terminates in quadrant II,
The reference angle = $ 180^{\circ} - 120^{\circ}$ = $60^{\circ}$
As $ 120^{\circ}$ ( $ 480^{\circ}$ also) terminates in quadrant II, its $\cot$ will be negative. Therefore by reference angle theorem-
$\cot 120^{\circ}$ = - $\cot 60^{\circ}$
= - $\frac{\sqrt 3}{3}$
Combining all the above, we get-
$\cot 480^{\circ}$ = $\cot 120^{\circ}$ = - $\cot 60^{\circ}$ = - $\frac{\sqrt 3}{3}$
