Question

Find the exact value of cot 330° in simplest form with a rational denominator.

Answer 1

Answer:

$\cot 330^{\circ} = -\sqrt{3}$

Step-by-step explanation:

The cotangent function can be rewritten by trigonometric relations, that is:

$\cot 330^{\circ} = \frac{1}{\tan 330^{\circ}} = \frac{\cos 330^{\circ}}{\sin 330^{\circ}}$ (1)

By taking approach the periodicity properties of the cosine and sine function (both functions have a period of 360°), we use the following equivalencies:

$\sin 330^{\circ} = \sin (-30^{\circ}) = -\sin 30^{\circ}$ (2)

$\cos 330^{\circ} = \cos (-30^{\circ}) = \cos 30^{\circ}$ (3)

By (2) and (3) in (1), we have following expression:

$\cot 330^{\circ} = -\frac{\cos 30^{\circ}}{\sin 30^{\circ}}$

If we know that $\sin 30^{\circ} = \frac{1}{2}$ and $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$, then the result of the trigonometric expression is:

$\cot 330^{\circ} = -\frac{\frac{\sqrt{3}}{2} }{\frac{1}{2} }$

$\cot 330^{\circ} = -\sqrt{3}$