Answer
(a) $ \frac{4}{5}$
(b) $ \frac{3}{5}$
(c) $ \sqrt {\frac{5+2\sqrt 5}{10}}$
(d) $ -\sqrt {\frac{5-2\sqrt 5}{10}}$
Work Step by Step
Given $tan\theta=\frac{1}{2}$ and $\pi\lt \theta\lt \frac{3\pi}{2}$, we have $sin\theta=-\frac{\sqrt 5}{5},cos\theta=-\frac{2\sqrt 5}{5}$ and $\frac{\pi}{2}\lt \frac{\theta}{2}\lt \frac{3\pi}{4}$.
(a) $sin(2\theta)=2sin\theta cos\theta=2(-\frac{\sqrt 5}{5})(-\frac{2\sqrt 5}{5})=\frac{4}{5}$
(b) $cos(2\theta)=1-2sin^2(\theta)=1-2(-\frac{\sqrt 5}{5})^2=\frac{3}{5}$
(c) $sin(\frac{\theta}{2})=\sqrt {\frac{1-cos\theta}{2}}=\sqrt {\frac{1+\frac{2\sqrt 5}{5}}{2}}=\sqrt {\frac{5+2\sqrt 5}{10}}$
(d) $cos(\frac{\theta}{2})=-\sqrt {\frac{1+cos\theta}{2}}=-\sqrt {\frac{1-\frac{2\sqrt 5}{5}}{2}}=-\sqrt {\frac{5-2\sqrt 5}{10}}$
