Question

examples
the captain of a small boat is delivering supplies to two lighthouses, as shown. his compass indicates that the lighthouse to his left is located at n30°w and the lighthouse to his right is located at n50°e. determine the compass direction he must follow when he leaves lighthouse b for lighthouse a.

Explanation:

Step1: Find angles in the triangle

In the triangle formed by the boat and the two lighthouses, we know one angle at the boat is \(30^{\circ}+ 50^{\circ}=80^{\circ}\). Let's call the angle at lighthouse \(A\) as \(A\) and at lighthouse \(B\) as \(B\). Using the sine - law \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(c = 12\) km, \(a = 9\) km and \(C = 80^{\circ}\). First, from the sine - law \(\frac{a}{\sin A}=\frac{c}{\sin C}\), so \(\sin A=\frac{a\sin C}{c}\).
Substitute \(a = 9\), \(c = 12\) and \(C = 80^{\circ}\) (so \(\sin C=\sin80^{\circ}\approx0.9848\)):
\(\sin A=\frac{9\times\sin80^{\circ}}{12}=\frac{9\times0.9848}{12}\approx0.7386\). Then \(A=\sin^{- 1}(0.7386)\approx47.6^{\circ}\).

Step2: Find angle \(B\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(B = 180^{\circ}-A - C\). Substitute \(A\approx47.6^{\circ}\) and \(C = 80^{\circ}\), we get \(B=180^{\circ}-47.6^{\circ}-80^{\circ}=52.4^{\circ}\).

Step3: Determine the compass direction

The direction from lighthouse \(B\) to lighthouse \(A\):
The angle measured from the north - south line at lighthouse \(B\). The angle between the line \(BA\) and the north - south line at \(B\) is \(50^{\circ}-47.6^{\circ}=2.4^{\circ}\) west of north. So the compass direction is \(N2.4^{\circ}W\).

Answer:

\(N2.4^{\circ}W\)