Question

A boat sails 3 km South, then 10 km Southeast. Use trigonometry to find the boat's distance and bearing from its starting point.

Answer 1

Answer on Question#38391, Math, Vector Calculus

The picture of movement looks approximately like this:

$b = 3km$ ， $c = 10km$ ， $a - ?$

The angle between $b$ and $c$ is 45 degrees. Hence, let us use cosine theorem to calculate $a$ :

$a^2 = b^2 + c^2 - 2bc\sin(45) = 3^2 + 10^2 - 2 \cdot 3 \cdot 10 \cdot \frac{1}{\sqrt{2}} = 109 - 30\sqrt{2}$ . Thus, $a = \sqrt{109 - 30\sqrt{2}} \approx 8.159km$ .