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

In ΔABC given a=5, b=7 & c= 4.  If ∡CAB = α then cot ((1/2)α)=?

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

Answered by liberty last updated on 22/Jun/21

cot ((1/2)α) = (√((s(s−a))/((s−b)(s−c))))  where s = ((a+b+c)/2)=((5+7+4)/2)=8  then cot ((1/2)α)=(√((8(8−5))/((8−7)(8−4))))                                = (√((8.3)/(1.4))) =(√6)

$$\mathrm{cot}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\alpha\right)\:=\:\sqrt{\frac{\mathrm{s}\left(\mathrm{s}−\mathrm{a}\right)}{\left(\mathrm{s}−\mathrm{b}\right)\left(\mathrm{s}−\mathrm{c}\right)}} \\ $$ $$\mathrm{where}\:\mathrm{s}\:=\:\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{2}}=\frac{\mathrm{5}+\mathrm{7}+\mathrm{4}}{\mathrm{2}}=\mathrm{8} \\ $$ $$\mathrm{then}\:\mathrm{cot}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\alpha\right)=\sqrt{\frac{\mathrm{8}\left(\mathrm{8}−\mathrm{5}\right)}{\left(\mathrm{8}−\mathrm{7}\right)\left(\mathrm{8}−\mathrm{4}\right)}} \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\sqrt{\frac{\mathrm{8}.\mathrm{3}}{\mathrm{1}.\mathrm{4}}}\:=\sqrt{\mathrm{6}}\: \\ $$

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