Math  /  Trigonometry

QuestionWhat is the exact value for cot(165)\cot \left(165^{\circ}\right) ?

Studdy Solution
Calculate cot(165)\cot(165^\circ) as the reciprocal of tan(165)\tan(165^\circ):
cot(165)=1tan(165)=1(23)=132\cot(165^\circ) = \frac{1}{\tan(165^\circ)} = \frac{1}{-(2 - \sqrt{3})} = \frac{1}{\sqrt{3} - 2}
Rationalize the denominator:
cot(165)=1323+23+2=3+2(32)(3+2)=3+234=3+21=(3+2)\cot(165^\circ) = \frac{1}{\sqrt{3} - 2} \cdot \frac{\sqrt{3} + 2}{\sqrt{3} + 2} = \frac{\sqrt{3} + 2}{(\sqrt{3} - 2)(\sqrt{3} + 2)} = \frac{\sqrt{3} + 2}{3 - 4} = \frac{\sqrt{3} + 2}{-1} = -(\sqrt{3} + 2)
Thus, the exact value is:
32\boxed{-\sqrt{3} - 2}

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