Answer:
[tex]\cot 330^{\circ} = -\sqrt{3}[/tex]
Step-by-step explanation:
The cotangent function can be rewritten by trigonometric relations, that is:
[tex]\cot 330^{\circ} = \frac{1}{\tan 330^{\circ}} = \frac{\cos 330^{\circ}}{\sin 330^{\circ}}[/tex] (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:
[tex]\sin 330^{\circ} = \sin (-30^{\circ}) = -\sin 30^{\circ}[/tex] (2)
[tex]\cos 330^{\circ} = \cos (-30^{\circ}) = \cos 30^{\circ}[/tex] (3)
By (2) and (3) in (1), we have following expression:
[tex]\cot 330^{\circ} = -\frac{\cos 30^{\circ}}{\sin 30^{\circ}}[/tex]
If we know that [tex]\sin 30^{\circ} = \frac{1}{2}[/tex] and [tex]\cos 30^{\circ} = \frac{\sqrt{3}}{2}[/tex], then the result of the trigonometric expression is:
[tex]\cot 330^{\circ} = -\frac{\frac{\sqrt{3}}{2} }{\frac{1}{2} }[/tex]
[tex]\cot 330^{\circ} = -\sqrt{3}[/tex]
