Question

Вычислить..........​

Answer 1

Используем 3 формулы:

$\sin a-\sin b=2\sin\frac{a-b}{2}\cos\frac{a+b}{2}$,

$\cos a+\cos b=2\cos\frac{a+b}{2}\cos\frac{a-b}{2}$,

$\cos a-\cos b=-2\sin\frac{a+b}{2}\sin\frac{a-b}{2}$.

$1)(...)=2\cos\frac{105\textdegree+75\textdegree}{2}\cos\frac{105\textdegree-75\textdegree}{2}=2\cos90\textdegree\cos15\textdegree=2\cdot0\cos15\textdegree=0$

$2)(...)=2\sin\frac{105\textdegree-75\textdegree}{2}\cos\frac{105\textdegree+75\textdegree}{2}=2\sin15\textdegree\cos90\textdegree=2\sin15\textdegree\cdot0=0$

$3)(...)=-2\sin\frac{\frac{11\pi}{12}+\frac{5\pi}{12} }{2}\sin\frac{\frac{11\pi}{12}-\frac{5\pi}{12}}{2}=-2\sin\frac{\frac{16\pi}{12} }{2}\sin\frac{\frac{6\pi}{12}}{2}=-2\sin\frac{16\pi}{24}\sin\frac{6\pi}{24}=-2\sin\frac{2\pi}{3}\sin\frac{\pi}{4}=-2\cdot\frac{\sqrt3}{2}\cdot\frac{\sqrt2}{2}=-\frac{\sqrt6}{2}.$

ОТВЕТ: 1) 0; 2) 0; 3) $-\frac{\sqrt6}{2}$

Answer 2

$1)\ \ cos105^\circ+cos75^\circ =cos(180^\circ-75^\circ)+cos75^\circ =-cos75^\circ +cos75^\circ =0\\\\\\2)\ \ sin105^\circ -sin75^\circ =sin(180^\circ-75^\circ)-sin75^\circ =sin75^\circ -sin75^\circ =0$

$3)\ \ cos\frac{11\pi}{12}-cos\dfrac{5\pi}{12}=cos\Big(\pi -\dfrac{\pi}{12}\Big)-cos\dfrac{5\pi}{12}=\Big[\; \dfrac{\pi}{12}=15^\circ \ ,\ \dfrac{5\pi}{12}=75^\circ \; \Big]=\\\\\\=-cos\dfrac{\pi}{12}-cos\dfrac{5\pi}{12}=-cos15^\circ -cos75^\circ =-2\cdot cos\dfrac{15^\circ +75^\circ}{2}\cdot cos\dfrac{75^\circ -15^\circ }{2}=\\\\\\=-2\cdot cos45^\circ \cdot cos30^\circ =-2\cdot \dfrac{\sqrt2}{2}\cdot \dfrac{\sqrt3}{2}=-\dfrac{\sqrt6}{2}$