Q.17 In \(\triangle ABC\) if \(a = 13\), \(b = 14\), \(c = 15\) then find the values of (i) \(\sec A\) (ii) \(\operatorname{cosec}\dfrac{A}{2}\).
Best answer
Answer: (i) \(\sec A = \dfrac53\) (ii) \(\operatorname{cosec}\dfrac A2 = \sqrt5\)
(i) \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc} = \dfrac{196 + 225 - 169}{2(14)(15)} = \dfrac{252}{420} = \dfrac35\), so \(\sec A = \dfrac53\).
(ii) \(s = \dfrac{13 + 14 + 15}{2} = 21\), \(s - b = 7\), \(s - c = 6\).
\(\sin\dfrac A2 = \sqrt{\dfrac{(s - b)(s - c)}{bc}} = \sqrt{\dfrac{7 \times 6}{14 \times 15}} = \sqrt{\dfrac15} = \dfrac{1}{\sqrt5}\), so \(\operatorname{cosec}\dfrac A2 = \sqrt5\).
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