Math Snap

STEP 1

1. The angle $\frac{\gamma}{2}$ is in the first quadrant, where all trigonometric functions are positive.
2. We are given $\cos \gamma = \frac{161}{289}$.
3. We need to use the half-angle identity for sine to find $\sin \left(\frac{\gamma}{2}\right)$.

STEP 2

1. Use the Pythagorean identity to find $\sin \gamma$.
2. Use the half-angle identity for sine to find $\sin \left(\frac{\gamma}{2}\right)$.

STEP 3

First, use the Pythagorean identity to find $\sin \gamma$. The identity is:
$$\sin^2 \gamma + \cos^2 \gamma = 1$$ Substitute $\cos \gamma = \frac{161}{289}$ into the identity:
$$\sin^2 \gamma + \left(\frac{161}{289}\right)^2 = 1$$ Calculate $\left(\frac{161}{289}\right)^2$:
$$\left(\frac{161}{289}\right)^2 = \frac{161^2}{289^2} = \frac{25921}{83521}$$ Substitute back into the identity:
$$\sin^2 \gamma + \frac{25921}{83521} = 1$$ Solve for $\sin^2 \gamma$:
$$\sin^2 \gamma = 1 - \frac{25921}{83521}$$ $$\sin^2 \gamma = \frac{83521 - 25921}{83521}$$ $$\sin^2 \gamma = \frac{57600}{83521}$$ Take the square root to find $\sin \gamma$:
$$\sin \gamma = \sqrt{\frac{57600}{83521}}$$ $$\sin \gamma = \frac{240}{289}$$

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:
$$\boxed{\frac{8}{17}}$$