Формулы двойного и половинного аргумента. Задача 31math100admin44242025-03-23T21:01:32+03:00
Задача 31. Вычислите \(\dfrac{{\sqrt 3 }}{2}\left( {\sin {{75}^ \circ }-\sin {{15}^ \circ }} \right)\left( {\sin {{75}^ \circ } + \sin {{15}^ \circ }} \right)\)
Решение
\(\dfrac{{\sqrt 3 }}{2}\left( {\sin {{75}^ \circ }-\sin {{15}^ \circ }} \right)\left( {\sin {{75}^ \circ } + \sin {{15}^ \circ }} \right) = \)
\( = \dfrac{{\sqrt 3 }}{2}\left( {\sin \left( {{{90}^ \circ }-{{15}^ \circ }} \right)-\sin {{15}^ \circ }} \right)\left( {\sin \left( {{{90}^ \circ }-{{15}^ \circ }} \right) + \sin {{15}^ \circ }} \right) = \)
\( = \dfrac{{\sqrt 3 }}{2}\left( {\cos {{15}^ \circ }-\sin {{15}^ \circ }} \right)\left( {\cos {{15}^ \circ } + \sin {{15}^ \circ }} \right) = \dfrac{{\sqrt 3 }}{2}\left( {{{\cos }^2}{{15}^ \circ }-{{\sin }^2}{{15}^ \circ }} \right) = \)
\( = \dfrac{{\sqrt 3 }}{2}\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2} \cdot \dfrac{{\sqrt 3 }}{2} = \dfrac{3}{4} = 0,75.\)
Ответ: 0,75.