Question

Помогите люди добрые​

Answer 1

$1)\ \ sin103^\circ 30'\cdot cos13^\circ 30'-cos103^\circ 30'\cdot sin13^\circ 30'=sin(103^\circ 30'-13^\circ 30')=\\\\=sin90^\circ =1\\\\\\2)\ \ cos53^\circ \cdot cos8^\circ +sin53^\circ \cdot sin8^\circ =cos(53^\circ -8^\circ )=cos45^\circ =\dfrac{\sqrt2}{2}$

$3)\ \ sin\dfrac{\pi}{12}\cdot cos\dfrac{\pi}{6}+cos\dfrac{\pi}{12}\cdot sin\dfrac{\pi}{6}=sin\Big(\dfrac{\pi}{12}+\dfrac{\pi}{6}\Big)=sin\dfrac{\pi}{4}=\dfrac{\sqrt2}{2}$

$4)\ \ cos\dfrac{\pi}{9}\cdot cos\dfrac{8\pi}{9}-sin\dfrac{\pi}{9}\cdot sin\dfrac{8\pi}{9}=cos\Big(\dfrac{\pi}{9}+\dfrac{8\pi}{9}\Big)=cos\pi =-1$

$5)\ \ cos15^\circ =cos(45^\circ -30^\circ )=cos45^\circ \cdot cos30^\circ +sin45^\circ \cdot sin30^\circ =\\\\=\dfrac{\sqrt2}{2}\cdot \dfrac{\sqrt3}{2}+\dfrac{\sqrt2}{2}\cdot \dfrac{1}{2}=\dfrac{\sqrt6+\sqrt2}{4}$

$6)\ \ cos135^\circ =cos(90^\circ +45^\circ )=cos90^\circ \cdot cos45^\circ -sin90^\circ \cdot sin45^\circ =\\\\=0\cdot \dfrac{\sqrt2}{2}-1\cdot \dfrac{\sqrt2}{2}=-\dfrac{\sqrt2}{2}$