a) 4⋅sin 12π⋅sin 125π=1
Czyli:
sin 12π⋅sin 125π=41
Zauważmy, że:
sin 12π⋅sin 125π=sin (123π−122π)⋅sin (123π+122π)=
=(sin 4π⋅cos 6π−cos 4π⋅sin 6π)⋅(sin 4π⋅cos 6π+cos 4π⋅sin 6π)=
=(22⋅23−22⋅21)(22⋅23+22⋅21)=
=(46−42)(46+42)=166−162=164=41
c.n.w.
b) sin 85π⋅sin 87π=42
Zauważmy, że:
sin 85π⋅sin 87π=sin (8π+84π)⋅sin (π−8π)=
=sin (8π+2π)⋅sin (π−8π)=cos 8π⋅sin 8π=
=21⋅(2⋅sin 8π⋅cos 8π)=21⋅sin(2⋅8π)=
=21⋅sin 4π=21⋅22=42=
c.n.w.
c) 2⋅cos 5π⋅cos 52π=21
Czyli:
cos 5π⋅cos 52π=41
Zauważmy, że:
cos 5π⋅cos 52π=cos 5π⋅cos 52π⋅sin 5πsin 5π=
=sin 5πcos 5π⋅cos 52π⋅sin 5π=sin 5π21⋅2⋅cos 5π⋅sin 5π⋅cos 52π=
=sin 5π21⋅sin 52π⋅cos 52π=sin 5π21⋅21⋅2⋅sin 52π⋅cos 52π=
=sin 5π41⋅sin 54π=sin 5π41⋅sin (π−5π)=sin 5π41⋅sin 5π=41
c.n.w.
d) cos 7π⋅cos 72π⋅cos 74π=−81
Zauważmy, że:
cos 7π⋅cos 72π⋅cos 74π=cos 7π⋅cos 72π⋅sin 72πsin 72π⋅cos 74π⋅sin 74πsin 74π=
=cos 7π⋅21⋅2⋅cos 72π⋅sin 72πsin 72π⋅21⋅2⋅cos 74π⋅sin 74πsin 74π=
=41⋅cos 7π⋅sin 72πsin 74π⋅sin 74πsin 78π=41⋅cos 7π⋅sin 72πsin 78π=
=41⋅cos 7π⋅sin 72πsin (π+7π)=41⋅cos 7π⋅sin 72π−sin 7π=
=−41⋅21⋅2⋅cos 7π⋅sin 72πsin 7π=−81⋅sin 72πsin 72π=−81
c.n.w.
