Question

100Б.Помогите решить задачу!
Преобразуйте в сумму тригонометрических функций произведенние: 2sin10° * cos15° * cos25°​

Answer 1

$2\sin10^\circ\cdot\cos15^\circ\cdot\cos25^\circ=2\cdot(\sin10^\circ\cdot\cos15^\circ)\cdot\cos25^\circ=$

$=2\cdot\dfrac{1}{2} (\sin(10^\circ+15^\circ)+\sin(10^\circ-15^\circ))\cdot\cos25^\circ=$

$=(\sin25^\circ+\sin(-5^\circ))\cdot\cos25^\circ=(\sin25^\circ-\sin5^\circ)\cdot\cos25^\circ=$

$=\sin25^\circ\cdot\cos25^\circ-\sin5^\circ\cdot\cos25^\circ=$

$=\dfrac{1}{2}( \sin(25^\circ+25^\circ)+\sin(25^\circ-25^\circ))-\dfrac{1}{2}( \sin(5^\circ+25^\circ)+\sin(5^\circ-25^\circ))=$

$=\dfrac{1}{2}( \sin50^\circ+\sin0^\circ)-\dfrac{1}{2}( \sin30^\circ+\sin(-20^\circ))=$

$=\dfrac{1}{2}( \sin50^\circ+0)-\dfrac{1}{2}\left( \dfrac{1}{2} -\sin20^\circ\right)=\dfrac{1}{2} \sin50^\circ+\dfrac{1}{2}\sin20^\circ\right-\dfrac{1}{4}$