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Geometry Level 2

If cotA \text{cotA} and cosecA \text{cosecA} are the roots of the quadratic equation 3 x 2 − 3 3 x + 2 = 0 \large 3x^2-3\sqrt{3}x+2=0

Find the value of cot ⁡ ( A 2 ) \large \cot(\frac{A}{2}) .

This is an original problem and belongs to the set My Creations

9 9 3 3 3\sqrt{3} 3 \sqrt{3} 3 3

Skanda Prasad by Skanda Prasad , 20, India

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1 solution

Skanda Prasad
Nov 9, 2017

We know, cot ⁡ A 2 = cotA + cosecA \cot\frac{A}{2}=\text{cotA}+\text{cosecA}

From Vieta's Formula, we have sum of roots that is cotA + cosecA = 3 \text{cotA}+\text{cosecA}=\sqrt{3}

Hence, cot ⁡ ( A 2 ) = 3 \cot(\frac{A}{2})=\boxed{\sqrt{3}}

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