Question

find the smallest angle of \\( \triangle abc \\).

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a = 123\), \(b\) be the side opposite \(\angle A\) (\(b = 35\)), and \(C = 100^{\circ}\).

Step2: Set up the proportion

\(\frac{35}{\sin B}=\frac{123}{\sin 100^{\circ}}\). Then \(\sin B=\frac{35\sin 100^{\circ}}{123}\).
Calculate \(\sin 100^{\circ}\approx0.9848\), so \(\sin B=\frac{35\times0.9848}{123}\approx\frac{34.468}{123}\approx0.2802\).
Then \(B\approx\sin^{- 1}(0.2802)\approx16.3^{\circ}\).

Step3: Find \(\angle A\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(\angle A=180^{\circ}-100^{\circ}-16.3^{\circ}=63.7^{\circ}\).

Answer:

\(\angle B\) (or \(B\))