Question

find the length of side c.
b = 58° a = 6.7 c = ?
round to the nearest tenth.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\).

Step2: Find angle \(C\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(C = 180^{\circ}-109^{\circ}-58^{\circ}=13^{\circ}\).

Step3: Substitute values into the Law of Sines formula

We know \(a = 6.7\), \(A=109^{\circ}\), \(C = 13^{\circ}\). So \(\frac{6.7}{\sin109^{\circ}}=\frac{c}{\sin13^{\circ}}\).

Step4: Solve for \(c\)

\(c=\frac{6.7\times\sin13^{\circ}}{\sin109^{\circ}}\).
Using a calculator, \(\sin109^{\circ}\approx0.946\), \(\sin13^{\circ}\approx0.225\).
\(c=\frac{6.7\times0.225}{0.946}=\frac{1.5075}{0.946}\approx1.6\)

Answer:

\(1.6\)