Question

Решить уравнение:
cos5x = sin7x​

Answer 1

$cos5x=sin7x$

$cos5x-sin7x=0$

$cos5x-cosleft(dfrac{pi}{2}-7xright)=0$

$-2sindfrac{5x+left(frac{pi}{2}-7xright)}{2}sindfrac{5x-left(frac{pi}{2}-7xright)}{2}=0$

$sindfrac{5x+frac{pi}{2}-7x}{2}sindfrac{5x-frac{pi}{2}+7x}{2}=0$

$sindfrac{frac{pi}{2}-2x}{2}sindfrac{12x-frac{pi}{2}}{2}=0$

$sinleft(dfrac{pi}{4}-xright)sinleft(6x-dfrac{pi}{4}right)=0$

$left[begin{array}{l} sinleft(dfrac{pi}{4}-xright)=0\ sinleft(6x-dfrac{pi}{4}right)=0end{array}$

$left[begin{array}{l} dfrac{pi}{4}-x=pi n\ 6x-dfrac{pi}{4}=pi nend{array}$

$left[begin{array}{l} x=dfrac{pi}{4}+pi n\ 6x=dfrac{pi}{4}+pi nend{array}$

$left[begin{array}{l} x=dfrac{pi}{4}+pi n, ninmathbb{Z}\ x=dfrac{pi}{24}+dfrac{pi n}{6}, ninmathbb{Z} end{array}$