Question

find the length of side c.
(diagram of triangle abc: angle at c is 152°, angle at b is 18°, side ac = 4.
find: c = ?, a = □°, a = □)

Explanation:

Step1: Find angle A

The sum of angles in a triangle is \(180^\circ\). So, \(A = 180^\circ - 152^\circ - 18^\circ\)
\(A = 10^\circ\)

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{c}{\sin C}=\frac{AC}{\sin B}\). Here, \(AC = 4\), \(B = 18^\circ\), \(C = 152^\circ\)
So, \(\frac{c}{\sin 152^\circ}=\frac{4}{\sin 18^\circ}\)
First, calculate \(\sin 152^\circ\) and \(\sin 18^\circ\). \(\sin 152^\circ=\sin(180^\circ - 28^\circ)=\sin 28^\circ\approx0.4695\), \(\sin 18^\circ\approx0.3090\)
Then, \(c=\frac{4\times\sin 152^\circ}{\sin 18^\circ}=\frac{4\times0.4695}{0.3090}\)
\(c=\frac{1.878}{0.3090}\approx6.08\) (rounded to two decimal places)

Answer:

The length of side \(c\) is approximately \(6.08\) (if we consider the more precise calculation, or using calculator for \(\sin\) values: \(\sin 152^\circ\approx0.46947\), \(\sin 18^\circ\approx0.30902\), then \(c = \frac{4\times0.46947}{0.30902}=\frac{1.87788}{0.30902}\approx6.08\)). If we round to a reasonable decimal place, it can also be presented as approximately \(6.1\) or more precisely calculated as above. But following the calculation steps, the value is approximately \(6.08\).