Question

$\sin 120°\cos 150°-\cos 240°\sin 330°$ is equal to

• A. $-\left(\frac{\sqrt{3}+1}{4}\right)$
• B. $-1$
• C. $1$
• D. $\frac{2}{3}$

Answer 1

The correct option is B

$-1$

Find the value of the expression:

$$\begin{gathered} =\sin 120°\cos 150°-\cos 240°\sin 330° \\ =\sin \left(180°-60°\right)\cos \left(180°-30°\right)-\cos \left(180°+60°\right)\sin \left(360°-30°\right) \\ =\sin 60°\left(-\cos 30°\right)-\left(-\cos 60°\right)\left(\sin 30°\right)[\because \sin (180°-\mathrm{\theta })=\mathrm{sin\theta },\cos (180°-\mathrm{\theta })=-\mathrm{cos\theta }, \\ \begin{array}{rcl}& & \cos \end{array}\begin{array}{rcl}& & (180°+\mathrm{\theta })\end{array}\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}-\mathrm{cos\theta },\sin (360°-\mathrm{\theta })& =& -\mathrm{sin\theta }]\end{array} \\ =\frac{\sqrt{3}}{2}\times -\frac{\sqrt{3}}{2}-\left(-\frac{1}{2}\right)\left(-\frac{1}{2}\right)[\because \sin 60°=\frac{\sqrt{3}}{2},\cos 30°=\frac{\sqrt{3}}{2},\cos 60°=\sin 30°=\frac{1}{2}] \\ =-\frac{3}{4}-\frac{1}{4} \\ =\frac{-4}{4} \\ =-1 \end{gathered}$$

Hence, the correct option is B.