Question

Прошу, помогите, задание прикреплено!

Answer 1

Ответ:

$\sin( \alpha + \beta ) = \\ = \sin( \alpha ) \cos( \beta ) + \cos( \alpha ) \sin( \beta )$

$\cos( \alpha ) < 0 \\ \sin( \beta ) < 0$

$\sin( \alpha ) = - \frac{1}{3} \\ \cos( \alpha ) = \sqrt{1 - \sin {}^{2} ( \alpha ) } \\ \cos( \alpha ) = - \sqrt{1 - \frac{1}{9} } = - \sqrt{ \frac{8}{9} } = - \frac{2 \sqrt{2} }{3}$

$\cos( \beta ) = - \frac{2}{3} \\ \sin( \beta ) = \sqrt{1 - \cos {}^{2} ( \beta ) } \\ \sin( \beta ) = - \sqrt{ 1 - \frac{4}{9}} = - \sqrt{ \frac{5}{9} } = - \frac{ \sqrt{5} }{3}$

$\sin( \alpha + \beta ) = - \frac{1}{3} \times ( - \frac{2}{3} ) + ( - \frac{2 \sqrt{2} }{3} ) \times ( - \frac{ \sqrt{5} }{3} ) = \\ = \frac{2}{9} + \frac{2 \sqrt{10} }{9} = \frac{2(1 + \sqrt{10} )}{9}$