Example 3 from Algebra and Trigonometry, 10.2 Non-right Triangles: Law of Cosines

Using the Law of Cosines to Solve a Communication Problem

On many cell phones with GPS, an approximate location can be given before the GPS signal is received. This is accomplished through a process called triangulation, which works by using the distances from two known points. Suppose there are two cell phone towers within range of a cell phone. The two towers are located 6000 feet apart along a straight highway, running east to west, and the cell phone is north of the highway. Based on the signal delay, it can be determined that the signal is 5,050 feet from the first tower and 2,420 feet from the second tower. Determine the position of the cell phone north and east of the first tower, and determine how far it is from the highway.

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
For simplicity, we start by drawing a diagram similar to Figure 6 and labeling our given information.

Figure 6

Using the Law of Cosines, we can solve for the angle $θ .$ Remember that the Law of Cosines uses the square of one side to find the cosine of the opposite angle. For this example, let $a = 2,420, b = 5,050,$ and $c = 6,000.$ Thus, $θ$ corresponds to the opposite side $a = 2,420.$

$\begin{matrix} a^{2} & = & b^{2} + c^{2} - 2 b c \cos θ \\ {(2, 420)}^{2} & = & {(5, 050)}^{2} + {(6, 000)}^{2} - 2 (5, 050) (6, 000) \cos θ \\ \begin{matrix} {(2, 420)}^{2} & - & {(5, 050)}^{2} - {(6, 000)}^{2} \end{matrix} & = & - 2 (5, 050) (6, 000) \cos θ \\ \frac{\begin{matrix} {(2, 420)}^{2} & - & {(5, 050)}^{2} - {(6, 000)}^{2} \end{matrix}}{- 2 (5, 050) (6, 000)} & = & \cos θ \\ \cos θ & \approx & 0.9183 \\ θ & \approx & \cos^{- 1} (0.9183) \\ θ & \approx & 23.3 ° \end{matrix}$
To answer the questions about the phone’s position north and east of the tower, and the distance to the highway, drop a perpendicular from the position of the cell phone, as in Figure 7. This forms two right triangles, although we only need the right triangle that includes the first tower for this problem.

Figure 7

Using the angle $θ = 23.3°$ and the basic trigonometric identities, we can find the solutions. Thus
$\begin{array}{l} \begin{array}{l} \cos (23.3°) = \frac{x}{5,050} \end{array} \\ x = 5,050 \cos (23.3°) \\ x \approx 4,638.15 feet \\ \sin (23.3°) = \frac{y}{5,050} \\ y = 5,050 \sin (23.3°) \\ y \approx 1,997.5 feet \end{array}$
The cell phone is approximately 4,638 feet east and 1998 feet north of the first tower, and 1998 feet from the highway.

How did it go?

Solution

Solution