M1 10. In \( \triangle \mathrm{UVW}, v=93 \mathrm{~cm}, u=49 \mathrm{~cm} \) and \( \angle \mathrm{U}=28^{\circ} \). Find all possible values of \( \angle \mathrm{V} \), to the nearest degree.

To find the possible values of \( \angle V \) in triangle \( UVW \), we can use the Law of Sines. According to the Law of Sines, \[ \frac{u}{\sin U} = \frac{v}{\sin V} \] Given \( u = 49 \, \text{cm} \), \( v = 93 \, \text{cm} \), and \( \angle U = 28^\circ \), we can rearrange the formula to find \( \sin V \): \[ \sin V = \frac{v \cdot \sin U}{u} = \frac{93 \cdot \sin(28^\circ)}{49} \] Calculating \( \sin(28^\circ) \) gives approximately \( 0.4695 \). So, \[ \sin V \approx \frac{93 \cdot 0.4695}{49} \approx \frac{43.5985}{49} \approx 0.8883 \] Now, we take the arcsine to find \( V \): \[ V \approx \arcsin(0.8883) \approx 62.7^\circ \] Since sine is positive in both the first and second quadrants, we also consider the second possible value: \[ \angle V' = 180^\circ - 62.7^\circ \approx 117.3^\circ \] Thus, rounding them to the nearest degree gives us two possible values for \( \angle V \): \[ \angle V \approx 63^\circ \quad \text{or} \quad \angle V \approx 117^\circ \] So, the possible values of \( \angle V \) to the nearest degree are \( 63^\circ \) and \( 117^\circ \).