Question

given the triangle
, find the length of side x using the law of sines. round your final answer to 4 decimal places.
x =

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180-(42 + 14)=124^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, if \(a = 12\), \(A = 42^{\circ}\), \(B=14^{\circ}\), and \(b = x\). So \(\frac{12}{\sin42^{\circ}}=\frac{x}{\sin14^{\circ}}\).

Step3: Solve for \(x\)

Cross - multiply: \(x=\frac{12\times\sin14^{\circ}}{\sin42^{\circ}}\).
We know that \(\sin14^{\circ}\approx0.2419\) and \(\sin42^{\circ}\approx0.6691\).
Then \(x=\frac{12\times0.2419}{0.6691}=\frac{2.9028}{0.6691}\approx4.3384\).

Answer:

\(4.3384\)