Question

COMPUTE :-

(I) cosec (13π/12)​

Answer 1

$\tt \huge{answer : } \\ \\ \sf \implies \:cosec( \frac{13\pi}{2} ) \\ \\ \sf \implies \:cosec(\pi + \frac{\pi}{12} ) \\ \\ \sf \implies \: - cosec( \frac{\pi}{12} ) \\ \\ \sf \implies \: - cosec \: 15 \: \degree \\ \\ \sf \implies \: \frac{ - 1}{sin \: 15 \degree} \\ \\ now \\ \\ \sf \implies \:sin \: 15 \degree \: = sin(45\degree - 30\degree) = sin \: 45\degree \times cos \: 45\degree - cos \: 45\degree \times sin \: 30\degree = \frac{1}{ \sqrt{2} } \times \frac{ \sqrt{3} }{2} - \frac{1}{ \sqrt{2} } \times \frac{1}{2} = \frac{ \sqrt{3} - 1 }{2 \sqrt{2} } \\ \\ now$

$\tt \implies \: cosec( \frac{13\pi}{12} ) = \frac{ - 1}{sin \: 15} = \frac{2 \sqrt{2} }{1 - \sqrt{3} } \times \frac{1 + \sqrt{3} }{1 + \sqrt{3} } = - \sqrt{2} - \sqrt{6}$