# Question #c3a08

Mar 24, 2017

$\Rightarrow 2 \sqrt{3} - 3$

#### Explanation:

$\frac{\sin {120}^{0} \times \csc {150}^{0} \times \tan {225}^{0}}{\sin {330}^{0} + \cos {210}^{0}}$

Assume that all angles are in degrees. Lets we find the values of the mentioned trigonometric ratios:-

$\sin 120 = S \in \left(180 - 60\right) = \sin 60 = \frac{\sqrt{3}}{2}$----(sine of an angle is positive in the second quadrant.)

$\csc 150 = \csc \left(180 - 30\right) = \csc 30 = 2$---(co-secant of an angle is also positive in second quadrant.)

$\tan 225 = \tan \left(180 + 45\right) = \tan 45 = 1$---(tangent of an angle is positive in third quadrant. )

$\sin 330 = \sin \left(360 - 30\right) = - \sin 30 = - \frac{1}{2}$---(sine of an angle is negative in fourth quadrant.)

$\cos 210 = \cos \left(180 + 30\right) = - \cos 30 = - \frac{\sqrt{3}}{2}$ ---(cosine of an angle is negative in third quadrant.)

$\Rightarrow \frac{\sin {120}^{0} \times \csc {150}^{0} \times \tan {225}^{0}}{\sin {330}^{0} + \cos {210}^{0}}$

$\Rightarrow \frac{\frac{\sqrt{3}}{2} \times 2 \times 1}{- \frac{1}{2} - \frac{\sqrt{3}}{2}}$

$\Rightarrow \frac{\sqrt{3}}{- \frac{1}{2} - \frac{\sqrt{3}}{2}}$

$\Rightarrow \frac{\sqrt{3}}{\frac{- 1 - \sqrt{3}}{2}}$

$\Rightarrow \frac{- 2 \sqrt{3}}{1 + \sqrt{3}}$

$\Rightarrow \frac{- 2 \sqrt{3}}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}}$

$\Rightarrow \frac{- 2 \sqrt{3} + 6}{1 - 3}$

$\Rightarrow \frac{- 2 \sqrt{3} + 6}{-} 2$

$\Rightarrow 2 \sqrt{3} - 3$