Answer
(a) $ -\frac{3}{5}$
(b) $ -\frac{4}{5}$
(c) $ \frac{1}{2}\sqrt {\frac{10-\sqrt {10}}{5}}$
(d) $ -\frac{1}{2}\sqrt {\frac{10+\sqrt {10}}{5}}$
Work Step by Step
Given $tan\theta=-3, sin\theta\lt0$, we know $\frac{3\pi}{2}\lt \theta\lt 2\pi$. We have $sin\theta=-\frac{3\sqrt {10}}{10}, cos\theta=\frac{\sqrt {10}}{10}$, and $\frac{3\pi}{4}\lt \frac{\theta}{2} \lt \pi$.
(a) $sin(2\theta)=2sin\theta cos\theta=2(-\frac{3\sqrt {10}}{10})(\frac{\sqrt {10}}{10})=-\frac{3}{5}$
(b) $cos(2\theta)=1-2sin^2(\theta)=1-2(-\frac{3\sqrt {10}}{10})^2=-\frac{4}{5}$
(c) $sin(\frac{\theta}{2})=\sqrt {\frac{1-cos\theta}{2}}=\sqrt {\frac{1-\frac{\sqrt {10}}{10}}{2}}=\frac{1}{2}\sqrt {\frac{10-\sqrt {10}}{5}}$
(d) $cos(\frac{\theta}{2})=-\sqrt {\frac{1+cos\theta}{2}}=-\sqrt {\frac{1+\frac{\sqrt {10}}{10}}{2}}=-\frac{1}{2}\sqrt {\frac{10+\sqrt {10}}{5}}$
