Question

If \, $\theta$ lies in 4th quadrant and $\sin \frac{\theta }{2}+\cos \frac{\theta }{2}=-\frac{1}{2}$ Then determine the value of $\sin 2\theta$

Answer 1

Squaring we have 1 + sin $\theta =\frac{1}{4}$ (Rule 34, P.324)
$\therefore sin\theta =-\frac{3}{4}$
$\therefore cos\theta =\sqrt{1-\frac{9}{16}}=+\frac{\sqrt{7}}{4}$ (4th quadrant)
$\therefore sin2\theta =2sin\theta cos\theta =-\frac{3\sqrt{7}}{8}$