In \( \triangle \mathrm{RST}, r=8.4 \mathrm{~cm}, t=8.1 \mathrm{~cm} \) and \( \angle \mathrm{T}=71^{\circ} \). Firld all possible values of \( \angle \mathrm{R} \), to the nearest 10 th of a degree.

To find the possible values of \( \angle R \) in triangle \( \triangle RST \) using the Law of Sines, we start with the known values: \( r = 8.4 \, \text{cm} \), \( t = 8.1 \, \text{cm} \), and \( \angle T = 71^\circ \). Using the Law of Sines: \[ \frac{r}{\sin R} = \frac{t}{\sin T} \] Substituting the known values: \[ \frac{8.4}{\sin R} = \frac{8.1}{\sin(71^\circ)} \] Now we can find \( \sin R \): \[ \sin R = \frac{8.4 \times \sin(71^\circ)}{8.1} \] Using \( \sin(71^\circ) \approx 0.9455 \): \[ \sin R \approx \frac{8.4 \times 0.9455}{8.1} \approx \frac{7.948}{8.1} \approx 0.980 \implies R \approx \arcsin(0.980) \] Calculating \( R \), \[ R \approx 78.4^\circ \] Now, since the sine function is positive in both the first and second quadrants, we can also find a second possible angle for \( R \): \[ R_{2} = 180^\circ - 78.4^\circ \approx 101.6^\circ \] Thus, the possible values for \( \angle R \), rounded to the nearest tenth of a degree, are approximately \( 78.4^\circ \) and \( 101.6^\circ \). So, \( \angle R \) can be \( 78.4^\circ \) or \( 101.6^\circ \).