Trig functions come in handy if you work for NASA or need to measure the vertical distance travelled by a rocket. In this example, a rocket is shot off and travels vertically as a scientist, who's a mile away, watches its flight. One second into the flight, the angle of elevation of the rocket is 30 degrees. Two seconds later, the angle of elevation is 60 degrees.

How far did the rocket travel in those two seconds? The figure shows the rocket rising vertically.

1. Identify the parts of the triangles that you can use to solve the problem.

In the figure, you see two right triangles. One is superimposed on the other and shares a side — the adjacent side. In both triangles, the relevant sides are those that are adjacent and opposite the angles of elevation.

2. Determine which trig function to use.

The ratio of the tangent uses the adjacent and opposite sides.

3. Write equations with the trig functions.

and

4. Solve for the values of x and y.

The tangents of 30-degree and 60-degree angles are convenient values.

If you refer to the Appendix, you see that

and

The value of y is the distance that the rocket traveled between the first and second sightings, so, solving for y, you get

The rocket rose about 1.155 miles in two seconds.