Find Height of Helicopter?

Question

Two observers, at the same time, measure the angle og elevation of a helicopter. One angle is measured as 35 degrees and the other angle as 50 degrees. If the observers are 200 feet apart and the helicopter lies over the line joining them, how high is the helicopter?

Answer 1

Draw triangle ABC
A is height of plane.
BC is ground level
BC = 200 ft
AD = h
AD is perpendicular to BC
Angle ABC = 50°
Angle ACB = 35°
BD = x ft
DC = y ft
x + y = 200

tan 50° = h / x
h = x tan 50 °
and
tan 35° = h / y
h = y tan 35°

x + y = 200
x = 200 - y

h = (200 - y) tan 50 °
h = y tan 35°

(200 - y) tan 50° = y tan 35°
200 tan 50° = y (tan 35° + tan 50°)
200 tan 50° = 1.89 y
y = 126.1
tan 35° = h / 123.1
h = 86.2 ft

Answer 2

Sketching the positions of the observers and the angles of elevations will give you a triangle with a base 200 ft with interior base angles of 35 and 50, designated Angle A and C respectively. The third interior angle will be the apex we shall name Angle B.

The sides opposite each angle will be a, b, and c respectively, so that b = 200 ft.
The helicopter would be at Angle B, and if you drop a vertical line right from under it perpendicular to side b, that vertical line would be the elevation of the helicopter, or "h".
Now "h" intersects side b at a point some distance from Angle C. Call that distance as "x" so that the distance from Angle A will then be 200 - x.
You will have formed 2 right triangles with a common leg "h", so that
tan A = h / (200-x) ---(Eqn 1), and
tan C = h / x, or x = h / tan C ----(Eqn. 2)

Do not be distracted by "x"; you won't have to solve for it, as you will see later.

Sustituting Eqn. 2 into Eqn. 1,
tan A = h / (200 - h / tan C)
= h / (200 tan C - h) / tan C
= h tan C / (200 tan C - h)
tan A (200 tan C - h) = h tan C
200 tan C tan A - h tan A = h tan C
200 tan C tan A = h tan C + h tan A
= h (tan C + tan A)
h = 200 tan C tan A / (tan C + tan A)
= 200 tan 50 tan 35 (tan 50 + tan A)
= 88.21 ft

Answer 3

Guess what? Its a triangle! If you have two angles, you have all three.

You have one side already... use the law of sines to find at least one more side.

Then use the basic trigonometry definition of sine.