if $sin\theta=\frac{4}{9},0lt\thetalt\frac{pi}{2}$, find the exact value of each of the following.\n(a)…

if $sin\theta=\frac{4}{9},0lt\thetalt\frac{pi}{2}$, find the exact value of each of the following.\n(a) $sin(2\theta)$ (b) $cos(2\theta)$ (c) $sin\frac{\theta}{2}$ (d) $cos\frac{\theta}{2}$\n(a) $sin(2\theta)=\frac{8sqrt{65}}{81}$\n(type an exact answer, using radicals as needed.)\n(b) $cos(2\theta)=\frac{49}{81}$\n(type an exact answer, using radicals as needed.)\n(c) $sin\frac{\theta}{2}=square$\n(type an exact answer, using radicals as needed.)

if $sin\theta=\frac{4}{9},0lt\thetalt\frac{pi}{2}$, find the exact value of each of the following.\n(a) $sin(2\theta)$ (b) $cos(2\theta)$ (c) $sin\frac{\theta}{2}$ (d) $cos\frac{\theta}{2}$\n(a) $sin(2\theta)=\frac{8sqrt{65}}{81}$\n(type an exact answer, using radicals as needed.)\n(b) $cos(2\theta)=\frac{49}{81}$\n(type an exact answer, using radicals as needed.)\n(c) $sin\frac{\theta}{2}=square$\n(type an exact answer, using radicals as needed.)

Answer

Explanation:

Step1: Find $\cos\theta$

Since $\sin^{2}\theta+\cos^{2}\theta = 1$, then $\cos\theta=\sqrt{1 - \sin^{2}\theta}$. Given $\sin\theta=\frac{4}{9}$, so $\cos\theta=\sqrt{1-\left(\frac{4}{9}\right)^{2}}=\sqrt{1-\frac{16}{81}}=\sqrt{\frac{81 - 16}{81}}=\sqrt{\frac{65}{81}}=\frac{\sqrt{65}}{9}$.

Step2: Use the half - angle formula for $\sin\frac{\theta}{2}$

The half - angle formula is $\sin\frac{\theta}{2}=\sqrt{\frac{1-\cos\theta}{2}}$. Substitute $\cos\theta=\frac{\sqrt{65}}{9}$ into the formula: $\sin\frac{\theta}{2}=\sqrt{\frac{1-\frac{\sqrt{65}}{9}}{2}}=\sqrt{\frac{\frac{9-\sqrt{65}}{9}}{2}}=\sqrt{\frac{9-\sqrt{65}}{18}}=\frac{\sqrt{9-\sqrt{65}}}{3\sqrt{2}}=\frac{\sqrt{18 - 2\sqrt{65}}}{6}$.

Answer:

$\frac{\sqrt{18 - 2\sqrt{65}}}{6}$