1. Вычислите: a) cos 75° б) cos40°cos10°+sin40°sin10° в) sin100°cos10°-cos100°sin10° г) cos(13π/3)

a) \(cos 75° = cos(45° + 30°) = cos45°cos30° - sin45°sin30° = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}\) б) Используем формулу \(cos(a+b) = cos(a)cos(b) - sin(a)sin(b)\) и \(cos(a-b) = cos(a)cos(b) + sin(a)sin(b)\). Тогда \(cos40°cos10° + sin40°sin10° = cos(40° - 10°) = cos30° = \frac{\sqrt{3}}{2}\) в) Используем формулу \(sin(a-b) = sin(a)cos(b) - cos(a)sin(b)\). Тогда \(sin100°cos10° - cos100°sin10° = sin(100° - 10°) = sin90° = 1\) г) \(cos(\frac{13π}{3}) = cos(\frac{12π + π}{3}) = cos(4π + \frac{π}{3}) = cos(\frac{π}{3}) = \frac{1}{2}\)