Question

a) ( mangle a=) 92,02 b) ( mangle b=) type your answer... c) ( mangle c=) type your answer...

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). For \(\angle B\), we can use the Law of Sines \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). We know \(a = 18\), \(b=13\), \(c = 12\) and \(m\angle A=92.02^{\circ}\).

Step2: Apply the Law of Sines for \(\angle B\)

From the Law of Sines \(\frac{\sin A}{BC}=\frac{\sin B}{AC}\). Substitute \(BC = 18\), \(AC=12\), \(\sin A=\sin(92.02^{\circ})\approx1\). Then \(\sin B=\frac{AC\times\sin A}{BC}=\frac{12\times\sin(92.02^{\circ})}{18}\approx\frac{12\times1}{18}=\frac{2}{3}\). So \(m\angle B=\sin^{- 1}(\frac{2}{3})\approx41.81^{\circ}\)

Step3: Use the angle - sum property of a triangle

Since the sum of angles in a triangle is \(180^{\circ}\), \(m\angle C=180^{\circ}-m\angle A - m\angle B\). Substitute \(m\angle A = 92.02^{\circ}\) and \(m\angle B\approx41.81^{\circ}\). Then \(m\angle C=180^{\circ}-92.02^{\circ}-41.81^{\circ}=46.17^{\circ}\)

Answer:

b) \(41.81\)
c) \(46.17\)