Question

The value of $\sin {20}^0\sin {40}^0\sin {80}^0$ is

• A. $\frac{1}{8}$
• B. $\frac{\sqrt{3}}{8}$
• C. $\frac{3}{8}$
• D. None

Answer 1

The correct option is B $\frac{\sqrt{3}}{8}$

$\sin {20}^o\sin {40}^o\sin {80}^o$

$=\sin {20}^o\sin (60-20{)}^o\sin (60+20{)}^o$

$=\sin {20}^o({\sin }^2{60}^{o}co{s}^2{20}^o-{\sin }^2{20}^{o}co{s}^2{60}^o)$

$=\sin {20}^o\left[{\sin }^2{60}^o\right(1-{\sin }^2{20}^o)-{\sin }^2{20}^o(1-{\sin }^2{60}^o\left)\right]$

$=\sin {20}^o[{\sin }^2{60}^o-{\sin }^2{20}^o-{\sin }^2{60}^o{\sin }^2{20}^o+{\sin }^2{20}^o{\sin }^2{60}^o]$

$=\sin {20}^o({\sin }^2{60}^o-{\sin }^2{20}^o)$

$=\sin {20}^o\left(\right(3/4)-{\sin }^2{20}^o)$

$=\frac{1}{4}(3\sin {20}^o-4{\sin }^3{20}^o)$

$=\frac{1}{4}\sin (3\times {20}^o)$

$=\frac{1}{4}\sin {60}^o$

$=\frac{\sqrt{3}}{8}$