Question

final exam
100 points possible answered: 19/22
question 18
you must show work here for full credit!
given the triangle (with a triangle image showing side 19, angles 12° and 30° and side x), find the length of side x using the law of sines. round your final answer to 4 decimal places.
x =
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Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). So the third angle \(A = 180^\circ - 12^\circ - 30^\circ = 138^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Here, let the side opposite \(12^\circ\) be \(19\), the side opposite \(30^\circ\) be \(x\), and the side opposite \(138^\circ\) be, say, \(a\) (not needed directly). So \(\frac{19}{\sin 12^\circ}=\frac{x}{\sin 30^\circ}\).

Step3: Solve for \(x\)

Rearrange the formula: \(x=\frac{19\times\sin 30^\circ}{\sin 12^\circ}\). Calculate \(\sin 30^\circ = 0.5\), \(\sin 12^\circ\approx0.2079\). Then \(x=\frac{19\times0.5}{0.2079}=\frac{9.5}{0.2079}\approx45.6950\).

Answer:

\(45.6950\)