Question

given the triangle
, find the length of side ( x ) using the law of sines. round your final answer to 4 decimal places.
( x = )
question help:
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Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180-(64 + 11)=105^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, if we consider \(a = 27\), \(A = 105^{\circ}\), \(B=11^{\circ}\) and \(b=x\). So \(\frac{27}{\sin105^{\circ}}=\frac{x}{\sin11^{\circ}}\).
Cross - multiply gives \(x=\frac{27\times\sin11^{\circ}}{\sin105^{\circ}}\).
We know that \(\sin105^{\circ}=\sin(60^{\circ}+ 45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659\) and \(\sin11^{\circ}\approx0.1908\).
Substitute the values: \(x=\frac{27\times0.1908}{0.9659}\).
Calculate \(27\times0.1908 = 5.1516\). Then \(x=\frac{5.1516}{0.9659}\approx5.3335\).

Answer:

\(5.3335\)