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Question Number 144164 by bobhans last updated on 22/Jun/21

$$\mathrm{In}\mathrm{\Delta }\mathrm{ABC}\mathrm{given}\mathrm{a}=5,\mathrm{b}=7\mathrm{\&}\mathrm{c}=4.$$ $$\mathrm{If}\measuredangle \mathrm{CAB}=\alpha \mathrm{then}\cot \left(\frac{1}{2}\alpha \right)=?$$

Answered by liberty last updated on 22/Jun/21

$$\cot \left(\frac{1}{2}\alpha \right)=\sqrt{\frac{\mathrm{s}(\mathrm{s}-\mathrm{a})}{(\mathrm{s}-\mathrm{b})(\mathrm{s}-\mathrm{c})}}$$ $$\mathrm{where}\mathrm{s}=\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2}=\frac{5+7+4}{2}=8$$ $$\mathrm{then}\cot \left(\frac{1}{2}\alpha \right)=\sqrt{\frac{8(8-5)}{(8-7)(8-4)}}$$ $$=\sqrt{\frac{8.3}{1.4}}=\sqrt{6}$$

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