Example 6 from Algebra and Trigonometry, 10.1 Non-right Triangles: Law of Sines

Finding an Altitude

Find the altitude of the aircraft in the problem introduced at the beginning of this section, shown in Figure 16. Round the altitude to the nearest tenth of a mile.

A diagram of a triangle where the vertices are the first ground station, the second ground station, and the airplane in the air between them. The angle between the first ground station and the plane is 15 degrees, and the angle between the second station and the airplane is 35 degrees. The side between the two stations is of length 20 miles. There is a dotted altitude line perpendicular to the ground side connecting the airplane vertex with the ground.

Figure 16

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
To find the elevation of the aircraft, we first find the distance from one station to the aircraft, such as the side $a,$ and then use right triangle relationships to find the height of the aircraft, $h .$
Because the angles in the triangle add up to 180 degrees, the unknown angle must be 180°−15°−35°=130°. This angle is opposite the side of length 20, allowing us to set up a Law of Sines relationship.
$\begin{array}{l} \begin{array}{l} \begin{array}{l} \end{array} \\ \frac{\sin (130°)}{20} = \frac{\sin (35°)}{a} \end{array} \\ a \sin (130°) = 20 \sin (35°) \\ a = \frac{20 \sin (35°)}{\sin (130°)} \\ a \approx 14.98 \end{array}$
The distance from one station to the aircraft is about 14.98 miles.

Now that we know $a,$ we can use right triangle relationships to solve for $h .$

$\begin{array}{l} \sin (15°) = \frac{opposite}{hypotenuse} \\ \sin (15°) = \frac{h}{a} \\ \sin (15°) = \frac{h}{14.98} \\ h = 14.98 \sin (15°) \\ h \approx 3.88 \end{array}$
The aircraft is at an altitude of approximately 3.9 miles.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

The diagram shown in Figure 17 represents the height of a blimp flying over a football stadium. Find the height of the blimp if the angle of elevation at the southern end zone, point A, is 70°, the angle of elevation from the northern end zone, point $B,$ is 62°, and the distance between the viewing points of the two end zones is 145 yards.

An oblique triangle formed from three vertices A, B, and C. Verticies A and B are points on the ground, and vertex C is the blimp in the air between them. The distance between A and B is 145 yards. The angle at vertex A is 70 degrees, and the angle at vertex B is 62 degrees.

Figure 17

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Figure 17

Solution

Your turn

Solution

Your turn