Вычислите:

а) cos 105 градусов б) sin 75 градусов в) cig 15 градусов

а)

cos105° = cos (60° + 45°) = cos60°*cos45° - sin60°*sin45° =

(1/2) * (√2) / 2 - (√3) / 2 * (√2) / 2 = (√2) / 4 (1-√3).

или cos105° = cos (90°+15°) = - sin15° = - √ (1-cos30°) / 2 = - √ (1-√3/2) / 2 =

- (1/2) √ (2-√3) = - (1/2) √ ((1-√3) ²/2) = - (1/2√2) (1-√3) = (√2) / 4 (1-√3).

б)

sin75° = sin (45° + 30°) = sin45°cos30° + cos45°sin30°=

1/√2 * (√3) / 2 + (1/√2) * (1/2) = (√3+1) / 2√2 = √2 (√3+1) / 4.

или sin75° = sin (90° - 15°) = cos15° = √ ((1+cos30°) / 2) = √ ((1 + (√3) / 2) / 2) =

(1/2) √ (2+√3) = (1/2) √ ((√3+1) ²*1/2) = (√3+1) / 2√2 = √2 (√3+1) / 4.

в)

ctq15° = ctq (45° - 30°) = (ctq45°ctq30° + 1) / (ctq30° - ctq45°) = (√3+1) / (√3 - 1) =

((√3+1) ²/2 = 2 + √3.

или ctq15° = √ ((1+cos30°) / (1-cos30°)) = √ (2+√3) / (2-√3) = 2+√3.