Theory question 110
Trigonometry(a) Without using Mathematical tables or calculators, simplify: (2 tan 60° + cos 30°)/sin 60°.
(b) From an aeroplane in the air and at a horizontal distance of 1050 m, the angle of depression of the top and base of a control tower at an instance are 36° and 41° respectively. Calculate, correct to the nearest metre, the: (i) height of the control tower; (ii) shortest distance between the aeroplane and the base of the control tower.
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(a) tan 60° = √3, cos 30° = sin 60° = √3/2
(2√3 + √3/2) ÷ √3/2 = (5√3/2) × (2/√3) = 5
(b)(i) Height of plane above the base = 1050 tan 41° ≈ 912.8 m
Height of plane above the top = 1050 tan 36° ≈ 762.9 m
Height of tower ≈ 912.8 − 762.9 ≈ 150 m
(ii) Distance = 1050/cos 41° ≈ 1391 m
The tower's height is the difference of two vertical distances from the plane.