Вычислите, используя формулы суммы/разности углов: 1) sin105o=___________________________________________________________; 2) cos15o=____________________________________________________________; 3) cos15o · sin15o=. Created by kamiltimergazin2005. algebra-ru

разности углов: 1) sin105o=____________________________________________________

kamiltimergazin2005 @kamiltimergazin2005

July 2022 1 6 Report

Вычислите, используя формулы суммы/разности углов:
1) sin105o=___________________________________________________________;
2) cos15o=____________________________________________________________;
3) cos15o · sin15o=

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Ответ:

$\displaystyle 1)\ \ sin105^\circ =sin(60^\circ +45^\circ )=sin60^\circ \cdot cos45^\circ +cos60^\circ \cdot sin45^\circ =\\\\=\frac{\sqrt3}{2}\cdot \frac{\sqrt2}{2}+\frac{1}{2}\cdot \frac{\sqrt2}{2}=\frac{\sqrt2\cdot (\sqrt3+1)}{4}=\frac{\sqrt6+\sqrt2}{4}$

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

$\star \ \ \ sin105^\circ =sin(90^\circ +15^\circ )=cos15^\circ \ \ \star \\\\\\3)\ \ cos15^\circ \cdot sin15^\circ =\dfrac{1}{2}\cdot sin30^\circ =\dfrac{1}{2}\cdot \dfrac{1}{2}=\dfrac{1}{4}$

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