Răspuns:
La rezolvarea exercitiilor vom aplica următoarele formule :
- sin a + sin b = 2 × sin (a+b)/2 × cos (a-b)/2
- sin a - sin b = 2 × cos (a+b)/2 × sin (a-b)/2
- cos a + cos b = 2 × cos (a+b)/2 × cos (a-b)/2
- cos a - cos b = - 2 × sin (a+b)/2 × sin (a-b)/2
a)
sin 72° + sin48° =
2sin [(72° + 48°)/2] × cos [(72° - 48°)/2] =
2sin60° × cos12° =
2 × (√3/2) × cos12° =
√3 × cos12°
b)
sin18° + sin72° =
2sin [(72° + 18°)/2] × cos [(72° - 18°)/2] =
2sin 90°/2 × cos 54°/2 =
2 sin 45° × cos 26° =
2 ×√2/2 × cos26° =
√2 × cos26°
c)
sin15° + sin75° =
sin(45° - 30°) + sin (45° + 30°) =
sin45°cos30° - sin30°cos45° + sin45°cos30° + sin30°cos45° =
2 sin45°cos30°=
2 ×√2/2 × √3/2 =
√2 × √3/2 =
√6 × 1/2
d)
2 cos ( 75° + 15°)/2 × cos (75° - 15°)/2 =
2 cos 90°/2 × cos 60°/2 =
2 cos 45° × cos 30° =
2 × √2/2 × 1/2 =
√2 × 1/2
e)
sin 105° - sin 15° =
2 cos ((105°+15°)/2) × sin ((105°-15°)/2) =
2 cos ((120°)/2) × sin ((90°)/2) =
2 cos 60° × sin 45° =
2 × 1/2 × √2/2 =
√2 × 1/2
f)
cos 165° - cos 15° =
- 2 sin (165° + 15°)/2 × sin (165° - 15°)/2 =
- 2 sin 180°/2 × sin 150°/2 =
- 2 sin 90° × sin 75° =
- 2 × sin75° =
- 2 × sin (45 ° + 30°) =
- 2 × (sin 45° cos 30° + cos 45° sin 30°) =
- 2 × (√2/2 × √3/2 + √2/2 × 1/2) =
- 2 × ( √6/4 + √2/4) =
- 2 × ( √6 + √2)/4
- (√6 + √2) × 1/2
