Задача 21705 Найдите значение выражения...
Условие
Решение
2π/17= α, тогда
4PI/17=2 α
8π/17=4 α
16π/17=8 α α
4cos α ·cos2 α ·cos4 α ·cos8 α =
Умножим и разделим на sin α
Тогда
sin α ·cos α =(1/2)sin2 α
sin2 α ·cos2 α =(1/2)sin4 α
sin4 α ·cos4 α =(1/2)sin8 α
sin8 α ·cos8 α =(1/2)sin16 α
4sin α ·cos α ·cos2 α ·cos4 α ·cos8 α /sin α =
=4·(1/2)sin2 α ·cos2 α ·cos4 α ·cos8 α /sin α =
=4·(1/2)·(1/2)sin4 α ·cos4 α ·cos8 α /sin α
=4·(1/2)·(1/2)·(1/2)sin8 α ·cos8 α /sin α=
=4·(1/2)·(1/2)·(1/2)·(1/2)sin16 α /sin α.
Итак,
4cos(2π/17)cos(4PI/17)cos(8π/17)cos(16π/17)=
=4·(1/2)·(1/2)·(1/2)·(1/2)sin32π|17/sin2π/17=
применяем формулы приведения
=(1/4)sin(2π–(2π/17))/sin(2π/17)=
=(1/4)(–sin(2π/17))/sin(2π/17)=–1/4