Question

1. solve △abc.

Explanation:

Step1: Find angle A

In a right triangle, the sum of angles is \(180^\circ\). Since \(\angle C = 90^\circ\) and \(\angle B = 65^\circ\), we have \(\angle A=180^\circ - 90^\circ - 65^\circ = 25^\circ\).

Step2: Find side BC (let's call it \(a\))

Using \(\tan B=\frac{AC}{BC}\), so \(BC = \frac{AC}{\tan B}\). Given \(AC = 15.5\space\text{cm}\) and \(\angle B = 65^\circ\), \(\tan65^\circ\approx2.1445\), then \(BC=\frac{15.5}{2.1445}\approx7.23\space\text{cm}\).

Step3: Find side AB (let's call it \(c\))

Using \(\cos B=\frac{BC}{AB}\), so \(AB=\frac{BC}{\cos B}\) or using \(\sin B=\frac{AC}{AB}\), so \(AB = \frac{AC}{\sin B}\). \(\sin65^\circ\approx0.9063\), then \(AB=\frac{15.5}{0.9063}\approx17.1\space\text{cm}\).

Answer:

\(\angle A = 25^\circ\), \(BC\approx7.23\space\text{cm}\), \(AB\approx17.1\space\text{cm}\)