Answer
- $\sqrt 3$
Work Step by Step
To find exact value of $\cot 510^{\circ}$, let's find its co-terminal angle between $0^{\circ}$ and $360^{\circ}$ first-
Co-terminal angle of $510^{\circ}$ = $510^{\circ} - 360^{\circ}$ = $150^{\circ}$
Therefore
$\cot 510^{\circ}$ = $\cot 150^{\circ}$
($510^{\circ}$ and $150^{\circ}$ are co-terminal and trigonometric functions of co-terminal angles are same)
Now to calculate exact value of $\cot 150^{\circ}$, let's find its reference angle first. As $ 150^{\circ}$ terminates in quadrant II,
The reference angle = $ 180^{\circ} - 150^{\circ}$ = $30^{\circ}$
As $ 150^{\circ}$ ( $ 510^{\circ}$ also) terminates in quadrant II, its $\cot$ will be negative. Therefore by reference angle theorem-
$\cot 150^{\circ}$ = - $\cot 30^{\circ}$
= - $\sqrt 3$
Combining all the above, we get-
$\cot 510^{\circ}$ = $\cot 150^{\circ}$ = - $\cot 30^{\circ}$ = - $\sqrt 3$
