Q.17 In \(\triangle ABC\), if \(a = 10\), \(b = 8\), \(c = 6\) then find the value of (i) \(\sin\left(\frac{C}{2}\right)\) (ii) \(\cos\left(\frac{B}{2}\right)\)
Best answer
Answer: (i) \(\sin\frac{C}{2} = \frac{1}{\sqrt{10}}\) (ii) \(\cos\frac{B}{2} = \frac{2}{\sqrt{5}}\)
\(s = \frac{a + b + c}{2} = \frac{10 + 8 + 6}{2} = 12\); \(s - a = 2\), \(s - b = 4\), \(s - c = 6\).
(i) \(\sin\frac{C}{2} = \sqrt{\frac{(s-a)(s-b)}{ab}} = \sqrt{\frac{2\times 4}{10\times 8}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}}\).
(ii) \(\cos\frac{B}{2} = \sqrt{\frac{s(s-b)}{ac}} = \sqrt{\frac{12\times 4}{10\times 6}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}\).
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