Question : If $\theta=135^{\circ} ; y=15^{\circ}$, what is the value of $2\cos(\theta)\sin(y)$?
Option 1: $\frac{\sqrt{3}-1}{2}$
Option 2: $\frac{1-\sqrt{3}}{2}$
Option 3: $2-\sqrt{3}$
Option 4: $\sqrt{3}-2$
Correct Answer: $\frac{1-\sqrt{3}}{2}$
Solution :
$\theta=135^{\circ}$
$y=15^{\circ}$
$2 \cos (\theta) \sin (y)$
= $2 \cos (135^{\circ}) \sin (15^{\circ})$
= $2 \cos (180^{\circ}-45^{\circ}) \sin (45^{\circ}-30^{\circ})$
= $2$ × $[-\cos45^{\circ}]$ × $[ \sin (45^{\circ})\cos(30^{\circ})-\cos (45^{\circ})\sin(30^{\circ})]$
= $2$ × $[ -\frac{1}{\sqrt2}]$ × $[ \frac{1}{\sqrt2}×\frac{\sqrt3}{2}-\frac{1}{\sqrt2}×\frac{1}{2}]$
= $2$ × $[-\frac{1}{\sqrt2}]$ × $[\frac{\sqrt3-1}{2\sqrt2}]$
= $\frac{1-\sqrt3}{2}$
Hence, the correct answer is $\frac{1-\sqrt3}{2}$.
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