Тема: "Преобразование произведения в сумму".
Три основные формулы, которыми будем пользоваться:
cosx*cosy=1/2(cos(x+y)+cos(x-y))
sinx*siny=1/2(cos(x-y)-cos(x+y))
sinx*cosy=1/2(sin(x+y)+sin(x-y))
sin82*30`*cos52*30`=1/2(sin(82*30`+52*30`)+sin(82*30`-52*30`))=
=1/2(sin135*+sin30*)=1/2sin135*+1/2*1/2=
=1/2sin(90*+45*)+1/4=1/2cos45*+1/4=![\frac{1}{2}*\frac{\sqrt{2}}{2}+\frac{1}{4}=\frac{\sqrt{2}+1}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%2A%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%2B%5Cfrac%7B1%7D%7B4%7D%3D%5Cfrac%7B%5Csqrt%7B2%7D%2B1%7D%7B4%7D)
sin82*30`*cos37*30`=1/2(sin(82*30`+37*30`)+sin(82*30`-37*30`)=
=1/2(sin120*+sin45*)=![\frac{1}{2}(\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2})=\frac{\sqrt{3}+\sqrt{2}}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%28%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%2B%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29%3D%5Cfrac%7B%5Csqrt%7B3%7D%2B%5Csqrt%7B2%7D%7D%7B4%7D)
cos37*30`*cos7*30`=1/2(cos(37*30`-7*30`)+cos(37*30`+7*30`)=
1/2(cos30*+cos45*)=![\frac{1}{2}(\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2})=</p> <p>=\frac{\sqrt{3}+\sqrt{2}}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%28%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%2B%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29%3D%3C%2Fp%3E%0A%3Cp%3E%3D%5Cfrac%7B%5Csqrt%7B3%7D%2B%5Csqrt%7B2%7D%7D%7B4%7D)
cos75*cos105*=1/2(cos(75*+105*)+cos(105*-75*))=1/2(cos180*+cos30*)=
![=\frac{1}{2}(-1+\frac{\sqrt{3}}{2})=\frac{\sqrt{3}-2}{4}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B2%7D%28-1%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%29%3D%5Cfrac%7B%5Csqrt%7B3%7D-2%7D%7B4%7D)
cos45*cos75*=1/2(cos(75*-45*)+cos(75*-45*))=1/2(cos120*+cos30*)=
![=\frac{1}{2}(-\frac{1}{2}+\frac{\sqrt{3}}{2})=\frac{\sqrt{3}-1}{4}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B2%7D%28-%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%29%3D%5Cfrac%7B%5Csqrt%7B3%7D-1%7D%7B4%7D)