Question

1. determine side a.

13.006 cm
63.7° 27.9°
a

Explanation:

Step1: Identify the Law to Use

We have a triangle with two angles and a side, so we use the Law of Sines. The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, where $a, b, c$ are sides and $A, B, C$ are the opposite angles.
First, find the third angle. The sum of angles in a triangle is $180^\circ$. Let the angles be $A = 63.7^\circ$, $B = 27.9^\circ$, so the third angle $C=180 - 63.7 - 27.9 = 88.4^\circ$. The side opposite angle $A$ (63.7°) is 13.006 cm, and side $a$ is opposite angle $B$ (27.9°).

Step2: Apply the Law of Sines

Using $\frac{a}{\sin B}=\frac{b}{\sin A}$, where $b = 13.006$ cm, $B = 27.9^\circ$, $A = 63.7^\circ$. So $a=\frac{b\sin B}{\sin A}$.
Calculate $\sin(27.9^\circ)\approx0.468$, $\sin(63.7^\circ)\approx0.897$.
Substitute values: $a=\frac{13.006\times0.468}{0.897}$.
First, $13.006\times0.468\approx6.087$. Then, $\frac{6.087}{0.897}\approx6.786$ cm.

Answer:

Approximately $6.79$ cm (or more precise $6.786$ cm)