1. **State the problem:** We have a spherical triangle with angles $A$, $B=90^\circ$, and $C=95^\circ$, and side $a=100^\circ$. We want to find angle $A$ using one of the given trigonometric relationships: - $\cos A = \cot C \cos a$ - $\cos A = \tan C \cos a$ - $\cos A = \cos C \sin a$ - $\cos A = \sin C \cos a$ 2. **Recall definitions:** - $\cot C = \frac{\cos C}{\sin C}$ - $\tan C = \frac{\sin C}{\cos C}$ 3. **Calculate trigonometric values for $C=95^\circ$ and $a=100^\circ$: ** - $\cos 95^\circ = \cos(90^\circ + 5^\circ) = -\sin 5^\circ \approx -0.0872$ - $\sin 95^\circ = \sin(90^\circ + 5^\circ) = \cos 5^\circ \approx 0.9962$ - $\cos 100^\circ = \cos(90^\circ + 10^\circ) = -\sin 10^\circ \approx -0.1736$ - $\sin 100^\circ = \sin(90^\circ + 10^\circ) = \cos 10^\circ \approx 0.9848$ 4. **Evaluate each expression for $\cos A$: ** - $\cos A = \cot C \cos a = \frac{\cos 95^\circ}{\sin 95^\circ} \times \cos 100^\circ = \frac{-0.0872}{0.9962} \times -0.1736 \approx 0.0152$ - $\cos A = \tan C \cos a = \frac{\sin 95^\circ}{\cos 95^\circ} \times \cos 100^\circ = \frac{0.9962}{-0.0872} \times -0.1736 \approx 1.982$ - $\cos A = \cos C \sin a = -0.0872 \times 0.9848 \approx -0.0859$ - $\cos A = \sin C \cos a = 0.9962 \times -0.1736 \approx -0.1729$ 5. **Check valid range for $\cos A$:** Since $\cos A$ must be between -1 and 1, all values are valid except $1.982$ which is invalid. 6. **Find $A$ for each valid $\cos A$: ** - For $\cos A = 0.0152$, $A = \arccos(0.0152) \approx 89.13^\circ$ - For $\cos A = -0.0859$, $A = \arccos(-0.0859) \approx 94.93^\circ$ - For $\cos A = -0.1729$, $A = \arccos(-0.1729) \approx 100.0^\circ$ 7. **Interpretation:** The formula $\cos A = \cot C \cos a$ gives $A \approx 89.13^\circ$, which is close to a right angle, consistent with $B=90^\circ$. The other formulas give different values. **Final answer:** Using $\cos A = \cot C \cos a$, angle $A \approx 89.13^\circ$.