Question

The value of $\text{sin12°sin48°sin54°}$ is :

• A. $\frac{2}{3}$
• B. $\frac{1}{2}$
• C. $\frac{1}{8}$
• D. $\frac{1}{3}$

Answer 1

The correct option is C

$\frac{1}{8}$

First we use the formula $\text{2sinAsinB=}\left[\text{cos(A-B)-cos(A+B)}\right]$

$\text{sin12°sin48°sin54°=}\frac{1}{2}\left[\text{cos(12-48)-cos(12+48)}\right]\text{sin54°}$

$$\begin{gathered} =\frac{1}{2}(\text{cos36°-cos60°)sin54°} \\ =\frac{1}{2}(\text{cos36°-}\frac{1}{2}\text{)sin54° (cos60°=}\frac{1}{2}\text{)} \\ =\frac{1}{2}\times \frac{1}{2}(2\text{cos36°sin54°-sin54°)} \\ =\frac{1}{4}(2\text{cos36°cos36°-cos36°) (since sin54°=sin(90}°\text{-36}°\text{)=cos36°)} \\ =\frac{1}{4}(2{\text{cos}}^2\text{36°-cos36°)} \\ =\frac{1}{4}(2{\left(\frac{\sqrt{5}+1}{4}\right)}^2-\left(\frac{\sqrt{5}+1}{4}\right))(\cos 36°=\frac{\sqrt{5}+1}{4}) \\ =\frac{1}{4}(2\left(\frac{5+1+2\sqrt{5}}{16}\right)-\left(\frac{\sqrt{5}+1}{4}\right)) \\ =\frac{1}{4}(2\left(\frac{6+2\sqrt{5}}{16}\right)-\left(\frac{\sqrt{5}+1}{4}\right)) \\ =\frac{1}{4}(\frac{1}{8}(6+2\sqrt{5})-\left(\frac{\sqrt{5}+1}{4}\right)) \\ =\frac{1}{4}(\frac{2}{8}(3+\sqrt{5})-\frac{\sqrt{5}+1}{4}) \\ =\frac{1}{4}(\frac{1}{4}(3+\sqrt{5}-\sqrt{5}-1)) \\ =\frac{1}{4}(\frac{1}{4}\left(2\right))=\frac{2}{16}=\frac{1}{8} \end{gathered}$$

Hence, option(C) is the correct answer.