Question

вычислите: sin 7п/24 • sin п/24 / cos 7п/8 • cos 5п/8
​

Answer 1

$\frac{ \sin( \frac{7\pi}{24} ) \sin( \frac{ \pi }{24} ) }{ \cos( \frac{7\pi}{8} ) \cos( \frac{5\pi}{8} ) } = \\ = \frac{ \frac{1}{2}( \cos( \frac{7\pi}{24} - \frac{\pi}{24} ) - \cos( \frac{7\pi}{24} + \frac{\pi}{24} ) )}{ \frac{1}{2} ( \cos( \frac{7\pi}{8} - \frac{5\pi}{8} ) - \cos( \frac{7\pi}{8} + \frac{5\pi}{8}) } = \\ = \frac{ \cos( \frac{6\pi}{24} ) - \cos( \frac{8\pi}{24} ) }{ \cos( \frac{2\pi}{8} ) - \cos( \frac{12\pi}{8} ) } = \\ = \frac{ \cos( \frac{\pi}{4} ) - \cos( \frac{\pi}{3} ) }{ \cos( \frac{\pi}{4} ) - \cos( \frac{3\pi}{2} ) } = \frac{ \frac{ \sqrt{2} }{2} - \frac{1}{2} }{ \frac{ \sqrt{2} }{2} - 0} = \\ = \frac{ \frac{ \sqrt{2} - 1}{2} }{ \frac{ \sqrt{2} }{2} } = \frac{2( \sqrt{2} - 1) }{2 \sqrt{2} } = \frac{ \sqrt{2} - 1}{ \sqrt{2} } = \\ = \frac{ \sqrt{2} ( \sqrt{2} - 1) }{ \sqrt{2} \sqrt{2} } = \frac{2 - \sqrt{2} }{2}$