Answer
(a) $ \frac{24}{25}$
(b) $ -\frac{7}{25}$
(c) $ \frac{\sqrt {5}}{5}$
(d) $ \frac{2\sqrt {5}}{5}$
Work Step by Step
Given $cos\theta=\frac{3}{5}$ and $0\lt \theta\lt \frac{\pi}{2}$, we have $sin\theta=\frac{4}{5}$.
(a) $sin(2\theta)=2sin\theta cos\theta=2(\frac{3}{5})(\frac{4}{5})=\frac{24}{25}$
(b) $cos(2\theta)=1-2sin^2(\theta)=1-2(\frac{4}{5})^2=-\frac{7}{25}$
(c) $sin(\frac{\theta}{2})=\sqrt {\frac{1-cos\theta}{2}}=\sqrt {\frac{1-3/5}{2}}=\frac{\sqrt {5}}{5}$
(d) $cos(\frac{\theta}{2})=\sqrt {\frac{1+cos\theta}{2}}=\sqrt {\frac{1+3/5}{2}}=\frac{2\sqrt {5}}{5}$
