Math trigonometry

Question

A helicopter is flying at an altitude of 475 yards. A building is located 180 yards from a point underneath the helicopter. Find the angle of depression from the helicopter to the building. Round to the nearest tenth.

Answer 1

A good way to start this problem is to draw a picture. We will define three points and then connect them to make a triangle which we will then use to solve the problem.

First step is to draw a coordinate plane.

1) Put your first point at the origin (0,0). This is the point directly below the helicopter.

2) Put your second point at (0,475). This point is the location of the helicopter.

3) Put your third point at (180,0), which is 180 yards away from the point underneath the helicopter.

If we now connect these three points, we create a triangle. Now let us define some angles. First lets note that the angle of depression from helicopter to building is equal to the angle of elevation from building to the helicopter. These are called alternate interior angles. For simplicity let us redefine this angle as x.

With our picture set up we can now use SinOppositeHypotenuse-CosAdjacentHypotenuse-TanOppositeAdjacent to solve for x. We use the trigonometric functions sine, cosine, and tangent to relate angles of a right triangle to the side lengths of that triangle. You can equate the tangent of an angle to the side opposite that angle divided by the side adjacent that angle. Lets apply that to the picture we have.

What is the tan(x)? The side opposite of x has a value of 475 yards. The side adjacent to x has a value of 180 yards.

Therefore, tan(x) = (475 yards / 180 yards). You can solve for the value of x using the inverse tangent function on your calculator. x = tan^-1(475/180) = 69.245. When rounding to the tenths place this angle is equal to 69.2 degrees.

Answer 2

From the description given, the points of interest together form a large right-triangle, with the helicopter at the apex.

The first sentence says that the helicopter is flying at an altitude of 475 yd, so the height of the right-triangle is 475 yd.

The second sentence says that the distance between a point underneath the helicopter and a building is 180 yd, so the base of the right-triangle is 180 yd.

And the third sentence asks us to find the "angle of depression" from the helicopter to the building, which we will interpret to mean the angle between the horizontal at the helicopters altitude and a line (of sight) from the helicopter to the building.

From geometry, we know that this horizontal is parallel to the ground, and if the line of sight forms and angle to one of these parallel lines, then the opposite interior angle will be have the same angle. This means that if we determine the angle between the ground and a line (of sight) from the building to the helicopter, it will be the same as the "angle of depression" from the helicopter to the building. Since these two angles have the same measure, we will let both of these angles be angle A.

Let A be the "angle of depression"; this angle can be calculated by recognizing that, for this right-triangle, height / base = tan A, so we have tan A = 475 yd / 180 yd = 2.638888... (approximately)

Now that we have the tangent of angle A, we need only take the arc-tangent to solve for A.

So, now we have A = arctan (475 yd / 180 yd) = 69.2 degrees (rounded to the nearest tenth).