Law of Sines: Ambiguous Case Score \( 0 / 1 \quad \) Penalty: 0.5 off Question In \( \Delta \leftrightarrows T \mathrm{TU}, t=2.4 \mathrm{~cm}, s=8.8 \mathrm{~cm} \) and \( \angle \mathrm{S}=153^{\circ} \). Find all possible values of \( \angle \mathrm{T} \), to the nearest 10th of \( a \) degree.

To solve for the possible values of \( \angle T \) in triangle \( \Delta STU \) using the Law of Sines, we start with the formula: \[ \frac{t}{\sin T} = \frac{s}{\sin S} \] Substituting the values \( t = 2.4 \) cm, \( s = 8.8 \) cm, and \( \angle S = 153^\circ \): \[ \frac{2.4}{\sin T} = \frac{8.8}{\sin(153^\circ)} \] First, we calculate \( \sin(153^\circ) \) which is approximately \( 0.8192 \): \[ \frac{2.4}{\sin T} = \frac{8.8}{0.8192} \] Calculating \( \frac{8.8}{0.8192} \): \[ \frac{8.8}{0.8192} \approx 10.73 \] Now, solving for \( \sin T \): \[ \sin T = \frac{2.4}{10.73} \approx 0.223 \] To find \( \angle T \), we take the arcsine: \[ T \approx \arcsin(0.223) \approx 12.8^\circ \] However, there's also the ambiguous case to consider since \( \sin(180^\circ - \theta) = \sin(\theta) \). Therefore, we have: \[ T' = 180^\circ - 12.8^\circ \approx 167.2^\circ \] Since \( T + S \) must be less than \( 180^\circ \) in a triangle, only \( T \approx 12.8^\circ \) is valid, while \( T' = 167.2^\circ \) remains invalid for this triangle configuration. So, the possible value of \( \angle T \) is approximately **12.8 degrees**.