1. **Problem 1:** Find the coordinates of point P on the unit circle for angle $A = \frac{5\pi}{6}$. The unit circle has radius 1, so coordinates for angle $\theta$ are $(\cos \theta, \sin \theta)$. 2. Calculate the coordinates for $A = \frac{5\pi}{6}$: $$\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}, \quad \sin \frac{5\pi}{6} = \frac{1}{2}$$ 3. Therefore, the coordinates are $\boxed{\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)}$. This matches **Option 1**. 4. **Problem 2:** Given point $(7,-1)$ on terminal side of angle $A$ in standard position, find $\cos A$. 5. Find the radius (distance from origin to point): $$r = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}$$ 6. The cosine is the adjacent side over hypotenuse, or $x/r$: $$\cos A = \frac{7}{5\sqrt{2}} = \frac{7\sqrt{2}}{10}$$ 7. So $\boxed{\frac{7\sqrt{2}}{10}}$, which matches **Option 1**. **Final answers:** - Coordinates for $A=\frac{5\pi}{6}$: $\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ - $\cos A$ for terminal side passing through $(7,-1)$ is $\frac{7\sqrt{2}}{10}$