§ 25: 25.33 (1-2)

25.33 Доведіть, що:

1) sin18°cos36° • 2cos18°2cos18°⁡= 2 sin36°cos36°4 • 18°= sin72°4 • 18° = cos18°4cos18°= 14;

2) 8sinπ9cosπ9cos2π9cos4π9sinπ9⁡– 4sin2π9cos2π9cos4π9sinπ9⁡= 2sin4π9cos4π9sinπ9⁡= sin8π9sinπ9 = sin(π- π9)sinπ9⁡= sinπ9sinπ9 = 1;

3) cosπ7 • cos4π7 • cos5π7 = 18.

Л. ч. = cosπ7 • cos4π7 • cos5π7 = 2sinπ 7 • cosπ7 • cos4π7 • cos5π72sinπ7⁡= sin2π7 • cos4π7 • cos5π72sinπ7⁡= sin2π7 • cos4π7 • cos(π- 2π7)2sinπ7⁡= -2sin2π7 • cos2π7 • cos4π72 • 2sinπ7⁡ = -2sin4π7 • cos4π72 • 4sinπ7⁡ = -sin8π78 • sinπ7 = -sin(π+ π7)8sinπ7⁡= sinπ7sinπ7 = 18 = П. ч.

Тобто cosπ7 • cos4π7 • cos5π7 = 18 – доведено.

4) sin6° • sin42° • cos12° • cos24° = 116.

Л.ч. sin6°• cos12° • cos24° • sin42° = 2sin6° • cos6° • cos12° • cos24° • sin42°2cos6° = 2sin24° • cos24° • sin42°2 • 4 cos6°⁡= sin48° • cos48°2 • 8cos6° = sin96°16cos6° = sin(90° + 6°)16cos6°⁡= 116 = П. ч.

Тобто sin6° • sin42° • cos12° • cos24° = 116 – доведено.