Question

In triangle ABC If a = 13, b = 14, c = 15. Find the value of
a) cos B
b) sin A/2
c) cos A/2
d) A(triangle ABC)​

Answer 1

$In \:\triangle ABC ,\\a = 13 , \: b = 14 , \: and \: c = 15$

$s = \frac{a+b+c}{2}\\= \frac{13+14+15}{2}\\=\frac{42}{2} \\= 21 \:---(1)$

$a ) Value \: of \: cos B \\= \frac{c^{2} + a^{2} - b^{2}}{2ca}$

$= \frac{ 15^{2} + 13^{2} - 14^{2}}{2\times 15 \times 13}$

$= \frac{ 225 + 169 - 196 }{30\times 13}$

$= \frac{294 - 196}{30\times 13}$

$= \frac{98}{15\times 13}\\=\frac{49}{195} \: --(2)$

$b ) Value \: of \: sin \:\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}} \\= \sqrt{\frac{(21-14)(21-15)}{14\times 15}} \\= \sqrt{\frac{7\times 6}{14\times 15}}\\= \frac{1}{5} \: ---(3)$

$c ) Value \: of \: Cos \:\frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}} \\= \sqrt{\frac{21(21-13)}{14\times 15}}\\= \sqrt{\frac{21\times 8}{14\times 15}}\\= \frac{4}{5} \: ---(4)$

$d ) Area \: \triangle ABC = \sqrt{ s(s-a)(s-b)(s-c) }\\= \sqrt{ 21(21-13)(21-14)(21-15)}\\= \sqrt{21\times 8\times 7\times 6 }\\= 84$

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