Question

Prove that sin120°.sin140°.sin160°=root 3/8

Answer 1

Prove that sin120° .sin140° .sin160° =3/8
Answer=
Note
that, sin120sin140sin160=sin20sin40sin60=sin602(cos20−cos60)sin⁡120sin⁡140sin⁡160=sin⁡20sin⁡40sin⁡60=sin⁡602(cos⁡20−cos⁡60)... (1)

sin60,cos60sin⁡60,cos⁡60 are well known values. We will try to get cos20cos⁡20.

Now, cos3x=4cos3x−3cosxcos⁡3x=4cos3⁡x−3cos⁡x. Substituting x=20x=20, we get,

4cos320−3cos20=124cos3⁡20−3cos⁡20=12

That is,

8cos320−6cos20−1=08cos3⁡20−6cos⁡20−1=0

Therefore, cos20cos⁡20 is the root of the equation 8x3−6x−1=08x3−6x−1=0. One can solve this cubic equation to find the value of cos20cos⁡20. Substitute it back in (1) to obtain the desired value.