Finding an angle from two spotlights in an aircraft

Key concept: The spotlight setup is a right-triangle trigonometry problem with a 10 m horizontal separation and a 20 m vertical drop.

Step by step

Identify the geometry

The nose light points straight down, so it already reaches the point 20 m below the aircraft. The second light is mounted on the tail, which is 10 m away along the body of the aircraft. That means the beam from the tail must travel from the tail position to the same point directly below the nose.

Because the target point is 20 m below the aircraft and 10 m forward from the tail light, the beam forms a right triangle. The vertical leg is 20 m and the horizontal leg is 10 m. The angle we want is the angle between the second light beam and the body of the aircraft, so it is the angle the beam makes with the horizontal aircraft body.

Set up the trigonometric ratio

For the right triangle, the tangent of the angle equals opposite divided by adjacent. Here, the opposite side is 20 m and the adjacent side is 10 m, so

$\tan \theta = \frac{20}{10} = 2.$

Therefore,

$\theta = \tan^{-1}(2).$

Using a calculator,

$\theta \approx 63.4^\circ.$

That is the angle the second spotlight must be set at, measured with respect to the body of the aircraft.

Why this works

The key idea is that the beam must meet the same point as the downward beam. Since the nose light hits the point directly below the aircraft, the tail light must compensate for the 10 m horizontal offset. Trigonometry converts that geometric offset into an angle.

You can also check the answer by thinking in terms of slope. A rise of 20 m over a run of 10 m gives a steep beam, and an angle above 45 degrees is expected. Since the vertical change is twice the horizontal change, an angle a little above 63 degrees is reasonable.

Common mistake to avoid

A frequent error is to use the angle with the vertical instead of the angle with the aircraft body. If you measure from the vertical, you would get the complementary angle, about 26.6 degrees. The question specifically asks for the angle with respect to the body of the aircraft, so the correct value is the larger angle, 63.4 degrees.

Final answer

The second light should be set at approximately 63.4° to the body of the aircraft.

Pitfall alert

A common trap in this aircraft-light problem is mixing up the reference angle. The beam is not being measured from the vertical line straight down; it is measured from the body of the aircraft, which acts like the horizontal side of the triangle. If you accidentally take the complementary angle, you will report about 26.6 degrees instead of the requested setting. Another mistake is to reverse the 20 m and 10 m sides in the tangent ratio. The tangent should compare the vertical drop to the horizontal separation, so the correct setup is tan theta = 20/10, not 10/20. A final check is physical sense: the tail light must point fairly steeply downward, because it needs to reach a point 20 m below while starting 10 m behind the nose beam.

Try different conditions

If the aircraft were still 20 m above the ground but the tail light were mounted 12 m behind the nose instead of 10 m, the same method would apply with a new right triangle. The angle would satisfy tan theta = 20/12, so theta = arctan(5/3), which is about 59.0 degrees. If the altitude changed to 30 m while the tail remained 10 m behind the nose, then tan theta = 30/10 = 3, giving theta about 71.6 degrees. These variants show that the angle depends on the ratio of vertical drop to horizontal separation, not on the absolute values alone.

Further reading

right triangle trigonometry, tangent ratio, angle of elevation and depression