Question

sin 100° · sin 120° · sin 140°. sin 160° (No calculator)
Step by step please or please tell how to.do​

Answer 1

$\bf\,sin\,100^{\circ}=sin(90^{\circ}+10^{\circ})=cos\,10^{\circ}sin100$

$\bf\,sin\,120^{\circ}=sin(90^{\circ}$

$\bf\,sin\,140^{\circ}=sin(90^{\circ}+50^{\circ})=cos\,50^{\circ}sin140$

$\bf\,sin\,160^{\circ}=sin(90^{\circ}+70^{\circ})=cos\,70^{\circ}sin160$

$\text{Now,}$

$sin\,100^{\circ}\;sin\,120^{\circ}\;sin\,140^{\circ}\;sin\,160^{\circ}sin100$

$=cos\,10^{\circ}(\frac{\sqrt3}{2})cos\,50^{\circ}\;cos\,70^{\circ}$

$=\frac{\sqrt3}{2}[cos\,50^{\circ}cos\,10^{\circ}\;cos\,70^{\circ}]$

$=\frac{\sqrt3}{2}[cos\,(60^{\circ}-10^{\circ})cos\,10^{\circ}\;cos\,(60^{\circ}+10^{\circ})]$

$\text{We know that,}$

$\boxed{\bf\,cos(60-A)\;cosA\;cos(60+A)=\frac{1}{4}cos\,3A}}$

$=\frac{\sqrt3}{2}[\frac{1}{4}cos3(10^{\circ})]$

$=\frac{\sqrt3}{2}[\frac{1}{4}cos30^{\circ}]$

$=\frac{\sqrt3}{2}[\frac{1}{4}\frac{\sqrt3}{2}]$

$=\frac{3}{16}$

$\therefore\bf\,sin\,100^{\circ}\;sin\,120^{\circ}\;sin\,140^{\circ}\;sin\,160^{\circ}=\frac{3}{16}$