QuestionSuppose that $\frac{\gamma}{2}$ is an angle in quadrant 1 and that $\cos \gamma=\frac{161}{289}$. Compute the exact value of $\sin \left(\frac{\gamma}{2}\right)$. $$\sin \left(\frac{\gamma}{2}\right)=23 \quad \times 0^{6} \frac{8}{17}$$

Studdy Solution
Use the half-angle identity for sine:
$$\sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{1 - \cos \gamma}{2}}$$
Substitute $\cos \gamma = \frac{161}{289}$:
$$\sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{1 - \frac{161}{289}}{2}}$$
Calculate $1 - \frac{161}{289}$:
$$1 - \frac{161}{289} = \frac{289 - 161}{289} = \frac{128}{289}$$
Substitute back into the half-angle identity:
$$\sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{\frac{128}{289}}{2}}$$ $$\sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{128}{578}}$$
Simplify the fraction:
$$\frac{128}{578} = \frac{64}{289}$$
Take the square root:
$$\sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{64}{289}}$$ $$\sin \left(\frac{\gamma}{2}\right) = \frac{8}{17}$$
The exact value of $\sin \left(\frac{\gamma}{2}\right)$ is:
$$\boxed{\frac{8}{17}}$$