1. **Problem Statement:** Given a spherical triangle with angles $A=90^\circ$, $B=100^\circ$, and $C=92^\circ$, find the side $b$ opposite angle $B$. 2. **Relevant Formula:** In spherical triangles, the Law of Cosines for sides states: $$\cos b = \cos a \cos c + \sin a \sin c \cos B$$ where $a$, $b$, and $c$ are sides opposite angles $A$, $B$, and $C$ respectively. 3. **Important Note:** Since $A=90^\circ$, side $a$ is $90^\circ$ (right angle), simplifying calculations. 4. **Using the Right-Angled Spherical Triangle Formula:** For a right-angled spherical triangle at $A$, the formula simplifies to: $$\cos b = \cos c \sin B$$ 5. **Calculate side $b$:** We need side $c$ opposite angle $C=92^\circ$. Using the spherical triangle angle sum property: $$A + B + C > 180^\circ$$ which is true here. 6. **Using the Law of Cosines for angles:** $$\cos C = -\cos A \cos B + \sin A \sin B \cos c$$ Substitute $A=90^\circ$: $$\cos 92^\circ = -\cos 90^\circ \cos 100^\circ + \sin 90^\circ \sin 100^\circ \cos c$$ Since $\cos 90^\circ=0$ and $\sin 90^\circ=1$: $$\cos 92^\circ = \sin 100^\circ \cos c$$ 7. **Calculate $\cos c$:** $$\cos c = \frac{\cos 92^\circ}{\sin 100^\circ}$$ Calculate values: $\cos 92^\circ \approx -0.0349$ $\sin 100^\circ \approx 0.9848$ So, $$\cos c \approx \frac{-0.0349}{0.9848} \approx -0.0354$$ 8. **Calculate $c$:** $$c = \arccos(-0.0354) \approx 92^\circ$$ 9. **Calculate $b$ using step 4 formula:** $$\cos b = \cos c \sin B$$ Calculate: $\cos 92^\circ \approx -0.0349$ $\sin 100^\circ \approx 0.9848$ So, $$\cos b = -0.0349 \times 0.9848 \approx -0.0344$$ 10. **Calculate $b$:** $$b = \arccos(-0.0344) \approx 92^\circ$$ **Final answer:** Side $b$ is approximately $92$ degrees and 1 minute.