Using the Law of Sines to solve for the unknown measures of \( \triangle A B C \) if: \( \angle A=35^{\circ}, \angle C=72^{\circ}, a=19 \) - \( m \angle B \) is 73 - \( b=73 \approx 73 \) - \( c=\square \) Round to two decimal places if needed. Assume \( \angle A \) is odposite side

To find the length of side \( c \) in triangle \( ABC \), we can first calculate angle \( B \). Knowing that the sum of angles in a triangle is \( 180^{\circ} \): \[ m \angle B = 180^{\circ} - (m \angle A + m \angle C) = 180^{\circ} - (35^{\circ} + 72^{\circ}) = 73^{\circ} \] Now, we can apply the Law of Sines: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] Substituting the known values: \[ \frac{19}{\sin 35^{\circ}} = \frac{b}{\sin 73^{\circ}} = \frac{c}{\sin 72^{\circ}} \] To find \( c \), we use: \[ \frac{c}{\sin 72^{\circ}} = \frac{19}{\sin 35^{\circ}} \] \[ c = \frac{19 \cdot \sin 72^{\circ}}{\sin 35^{\circ}} \] Calculating this gives: \[ c \approx \frac{19 \cdot 0.9563}{0.5736} \approx 31.99 \] So, rounding to two decimal places, \( c \approx 31.99 \). --- Did you know that the Law of Sines has its roots in ancient Greek geometry? Mathematicians like Hipparchus used early versions of it around 150 BC to chart stars and navigate the seas. This tool transformed the way we understand triangles, paving the way for astronomical calculations! In the real world, the Law of Sines isn’t just confined to classrooms. Engineers use it when designing structures to ensure everything fits together perfectly, and sailors use it for navigation by understanding their position relative to land and stars. Proving that math and geometry have a tangible impact on our daily lives!