Trigonometry: Height of a tower
David measured the angle of elevation
of the top of the tower from a point A to be 42◦
. He then moved 30 m
closer to the tower and from point B measured the angle of elevation to
the top of the tower to be 50◦
. To the nearest metre, determine the height
of the tower.
ABTˆ and T BCˆ form a straight angle. Therefore ABTˆ = 180◦ − 50◦ = 130◦
.
The angles in a triangle add to 180◦ so in 4T BA, AT Bˆ = 180◦ − 42◦ − 130◦ = 8◦ .
Let x represent the length of side BT and h represent T C, the required height.
Using the Sine Rule in 4ABT, x/sin 42◦ = 30/sin 8◦ and x = 30 sin 42◦/sin 8◦ ˙= 144.24.
Then in 4TBC, h/x = sin 50◦ and h = x sin 50 ˙= 110
The angles in a triangle add to 180◦ so in 4T BA, AT Bˆ = 180◦ − 42◦ − 130◦ = 8◦ .
Let x represent the length of side BT and h represent T C, the required height.
Using the Sine Rule in 4ABT, x/sin 42◦ = 30/sin 8◦ and x = 30 sin 42◦/sin 8◦ ˙= 144.24.
Then in 4TBC, h/x = sin 50◦ and h = x sin 50 ˙= 110
The height of the tower is 110 m.
