Question

what is the value of sin12°×sin48°×sin54°
plzzz solve it stepwise

Answer 1

Given sin12 * sin48 * sin54

= > 1/2(2 sin12 * sin48) * sin54

We know that 2sinasinb = cos(A - B) cos(A + B)

= > 1/2(cos(48 - 12) - cos(48 + 12)) * sin 54

= > 1/2(cos 36 - cos 60) * sin 54

= > 1/2(cos 36 - cos 60) * sin(90 - 36)

= > 1/2(cos 36 - cos 60) * cos 36

= > 1/2(cos 36 - 1/2) * cos 36

= > cos^36/2 - cos36/4

$= \ \textgreater \ (\frac{1 + \sqrt{5} }{ \frac{4}{2} })^2 - \frac{1 + \sqrt{5} }{4}$

$= \ \textgreater \ \frac{1 + \sqrt{5} + 2 \sqrt{5} }{32} - \frac{1 + \sqrt{5} }{16}$

$= \ \textgreater \ \frac{6 + 2\sqrt{5} }{32} - \frac{1 + \sqrt{5} }{16}$

$= \ \textgreater \ \frac{2(3 + \sqrt{5}) }{32} - \frac{1 + \sqrt{5} }{16}$

$= \ \textgreater \ \frac{3 + \sqrt{5} }{16} - \frac{1 + \sqrt{5} }{16}$

$= \ \textgreater \ \frac{3 + \sqrt{5} - 1 - \sqrt{5} }{16}$

= > 2/16

= > 1/8.


Hope this helps!

Answer 2

Hi,

Please see the attached file!


Thanks