Example 4 from Algebra and Trigonometry, 10.2 Non-right Triangles: Law of Cosines

Calculating Distance Traveled Using a SAS Triangle

Returning to our problem at the beginning of this section, suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles. How far from port is the boat? The diagram is repeated here in Figure 8.

A triangle whose vertices are the boat, the port, and the turning point of the boat. The side between the port and the turning point is 10 mi, and the side between the turning point and the boat is 8 miles. The side between the port and the turning point is extended in a straight dotted line. The angle between the dotted line and the 8 mile side is 20 degrees.

Figure 8

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The boat turned 20 degrees, so the obtuse angle of the non-right triangle is the supplemental angle, $180° - 20° = 160° .$ With this, we can utilize the Law of Cosines to find the missing side of the obtuse triangle—the distance of the boat to the port.
$\begin{array}{l} x^{2} = 8^{2} + 10^{2} - 2 (8) (10) \cos (160°) \\ x^{2} = 314.35 \\ x = \sqrt{314.35} \\ x \approx 17.7 miles \end{array}$
The boat is about 17.7 miles from port.

How did it go?

Solution

Solution