Sin50+sin70=?

sin27+cos63=?

cos15-cos75=?

cos (2 П/3) - cos (3 П/5) = ?

sin (П/12) - sin (5 П/12) = ?

*П-число Пи

1)

sin50+sin70 =

= 2 sin (50+70) / 2 * cos (50-70) / 2=

= 2 sin 60*cos10 = 2 * √3/2 cos10 = √3cos10

2)

sin27°+cos63° = sin (90°-63°) + cos63° = cos63° + cos63° = 2 cos63°

3)

cos15° - cos75° = - 2 sin (15°-75°) / 2 sin (15°+75°) / 2=

= - 2 sin (-30°) sin45° = 2*0,5*√2/2 = √2/2

4)

cos (2 П/3) - cos (3 П/5) = - 2 sin (2 П/3-3 П/5) / 2 sin (2 П/3+3 П/5) / 2=

= - 2 sin (П/15) / 2 sin (19 П/15) / 2=

= - 2 sinП/30 * sin19 П/30

Возможно в условии одинаковые знаменатели, т. е. нужно cos (2 П/5) вместо cos (2 П/3).

Тогда решение такое:

cos (2 П/5) - cos (3 П/5) =

= - 2 sin (2 П/5-3 П/5) / 2 sin (2 П/5+3 П/5) / 2=

= - 2 sin (-П/10) sinП/2=

= 2*√2/2 * sinП/10 = √2 / sinП/10

5)

sin (π/12) - sin5π/12 =

= 2sin[ (π/12-5π/12) / 2]*cos[ (π/12+5π/12) / 2] =

=2sin (-π/6) * cosπ/4 =

= - 2sinπ/6*cosπ/4 = - 2*1/2*√2/2 * = - √2/2