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 \(A = 180-(22 + 14)=144^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, let \(a = 30\), \(A = 22^{\circ}\), \(b=x\), \(B = 14^{\circ}\). So \(\frac{30}{\sin22^{\circ}}=\frac{x}{\sin14^{\circ}}\).

Step3: Solve for \(x\)

Cross - multiply to get \(x=\frac{30\times\sin14^{\circ}}{\sin22^{\circ}}\).
We know that \(\sin14^{\circ}\approx0.2419\) and \(\sin22^{\circ}\approx0.3746\).
Then \(x=\frac{30\times0.2419}{0.3746}=\frac{7.257}{0.3746}\approx19.3727\)

Answer:

\(19.3727\)