Question

1. find the missing parts of the triangle abc if ( b = 34^circ ), ( c = 82^circ ), and ( a = 5.6 ) cm. show all the details in a logical order.

Explanation:

Step1: Find angle A

In a triangle, the sum of interior angles is \(180^\circ\). So, \(A=180^\circ - B - C\).
Substitute \(B = 34^\circ\) and \(C = 82^\circ\):
\(A=180^\circ-(34^\circ + 82^\circ)=180^\circ - 116^\circ = 64^\circ\)

Step2: Find side b using the Law of Sines

Law of Sines: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
We know \(a = 5.6\) cm, \(A = 64^\circ\), \(B = 34^\circ\)
So, \(\frac{b}{\sin B}=\frac{a}{\sin A}\)
\(b=\frac{a\sin B}{\sin A}\)
\(\sin A=\sin64^\circ\approx0.8988\), \(\sin B=\sin34^\circ\approx0.5592\)
\(b=\frac{5.6\times0.5592}{0.8988}\approx\frac{3.13152}{0.8988}\approx3.48\) cm

Step3: Find side c using the Law of Sines

Using \(\frac{c}{\sin C}=\frac{a}{\sin A}\)
\(c=\frac{a\sin C}{\sin A}\)
\(\sin C=\sin82^\circ\approx0.9903\)
\(c=\frac{5.6\times0.9903}{0.8988}\approx\frac{5.54568}{0.8988}\approx6.17\) cm

Answer:

Angle \(A = 64^\circ\), side \(b\approx3.48\) cm, side \(c\approx6.17\) cm