Theory question 107
Trigonometry(a) The angles of depression of the top and bottom of a building are 51° and 62° respectively from the top of a tower 72m high. The base of the building is on the same horizontal level as the foot of the tower. Calculate the height of the building, correct to two significant figures.
(b) In the diagram, PR is a chord of the circle centre O and radius 30 cm. ∠POR = 120°. Calculate, correct to three significant figures. (i) the length of chord PR; (ii) the length of arc PQR; (iii) the perimenter of the shaded portion. (Take π = 3.142)
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(a) Horizontal distance d = 72/tan 62° = 72/1.8807 = 38.28 m.
Drop from the tower top to the building top = 38.28 × tan 51° = 38.28 × 1.2349 = 47.27 m.
Height of building = 72 − 47.27 = 24.7 ≈ 25 m.
(b)(i) PR = 2 × 30 sin 60° = 51.96 ≈ 52.0 cm.
(ii) Arc PQR (the 120° arc) = 120/360 × 2 × 3.142 × 30 = 62.84 ≈ 62.8 cm.
(iii) Perimeter of the shaded segment = 51.96 + 62.84 = 114.8 ≈ 115 cm.
Both angles of depression share the same horizontal distance between the tower and the building.