тригонометричен израз

[антонийо]

пресметнете израза
16*sin10*sin30 * sin 50 * sin 70

[123a]

Умножаваме и делим едновременно на $cos10$

[tex]8.\frac{2cos10.sin10.sin30.sin50.sin70}{cos10}[/tex]

$8.\frac{sin20.sin30.sin50.cos20}{cos10}$

$4.\frac{sin40.sin30.sin50}{cos10}$

$2.\frac{2.sin40.cos40.sin30}{cos10}$

$2.\frac{sin80.sin30}{cos10}=2.\frac{cos10.sin30}{cos10}=2.sin30=2.\frac{1}{2}=1$

Използвахме, че $ sin2\alpha=2sin\alpha.cos\alpha$ и $sin(90-\alpha)=cos\alpha$

[S.B.]

пресметнете израза
16*sin10*sin30 * sin 50 * sin 70

[tex]16.sin10.sin30.sin50.sin70 = 16.sin10.sin30.sin50.sin70.1[/tex]
$sin90 = 1$ за това изрзаът придобива вида:

$16.sin10.sin30.sin50.sin70.sin90 = 16 . sin10.sin50.sin70.[sin30.sin90] = $
$ = 16.sin10.sin50.sin70.\frac{1}{2}[cos(-60) - cos120] = $
$ = 16.sin10.sin50.sin70.\frac{1}{2}(\frac{1}{2} + \frac{1}{2}] = $
$ = 8.sin10.sin50.sin70 = $

$ = \frac{cos10}{cos10}.8.sin10.sin50.sin70 = $

$ = \frac{4.sin20.sin50.sin70}{cos10} = $

$ = \frac{4.sin20.sin(90 - 20).sin50}{cos10} = $

$ = \frac{4.sin20.cos20.sin50}{cos10} = $

$ = \frac{2.sin40.sin50}{cos10} = $

$ = \frac{2.sin40.sin(90 - 40)}{cos10} = $

$ = \frac{2.sin40.cos40}{cos10} = $

$ = \frac{sin80}{cos10} = \frac{sin80}{cos(90 - 80)} = \frac{sin80}{sin80} = 1$