Question

Sa se transforme in produs:
a)sin 72°+sin48°
b)sin18°+sin72°
c)sin15°+sin75°
d)cos15°+cos75°
e)sin  105° -sin 15°
f)cos 165°- cos 15°

Answer 1

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