Solving Word Problems Trigonometry :
Here we are going to see an example problem on the topic trigonometry.
Question 1 :
A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45° . The bird flies away horizontally in such away that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30° . Determine the speed at which the bird flies.(√3 = 1.732)
Let "x" be the speed at which the bird flies.
Time = distance / speed
2 = distance / x
Distance covered = 2x
In triangle ABC,
tan θ = Opposite side / Adjacent side
tan 30 = AB / BC
1/√3 = 80/ BC
BC = 80√3 ----(1)
tan 45 = AB / CD
1 = 80/(BC + CD)
1 = 80/(BC + 2x)
BC + 2x = 80
BC = 80 - 2x ------(2)
(1) = (2)
80√3 = 80 - 2x
By dividing the entire equation by 2, we get
40√3 = 40 - x
x = 40 - 40√3
x = 40(1 - 1.732)
x = 40 (-0.732)
x = -29.28
Since the speed will not be negative, speed of the bird is 29.28 m/s
Question 2 :
An aeroplane is flying parallel to the Earth’s surface at a speed of 175 m/sec and at a height of 600 m. The angle of elevation of the aeroplane from a point on the Earth’s surface is 37° at a given point. After what period of time does the angle of elevation increase to 53° ? (tan 53° = 1.3270, tan 37° = 0.7536)
Let "x" be the time taken by the aeroplane.
In triangle DBC,
tan 53 = DC/BC
1.3270 = 600/BC
BC = 600/1.320 ----(1)
In triangle ADC,
tan 37 = DC/AC
0.7536 = 600/(175x + BC)
175x + BC = 600/0.7536
BC = (600/0.7536) - 175x----(2)
(1) = (2)
600/1.3270 = (600/0.7536) - 175x
175x = (600/0.7536) - (600/1.3270)
175x = 600(1/0.7536 - 1/1.3270)
175x = 600(1.32 - 0.75)
175x = 600(0.57)
175x = 342
x = 342/175
x = 1.95 (approximately)
Question 3 :
A bird is flying from A towards B at an angle of 35° , a point 30 km away from A. At B it changes its course of flight and heads towards C on a bearing of 48° and distance 32 km away.
(i) How far is B to the North of A?
(ii) How far is B to the West of A?
(iii) How far is C to the North of B?
(iv) How far is C to the East of B?
(sin 55° = 0.8192, cos 55° = 0.5736, sin 42° = 0.6691, cos 42° = 0.7431)
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