(a) Without using Mathematical tables or calculators, simplify : \(\frac{2\tan 60° + \cos 30°}{\sin 60°}\) (b) From an aeroplane in the air and at a horizon...
Question 1 Report
(a) Without using Mathematical tables or calculators, simplify : \(\frac{2\tan 60° + \cos 30°}{\sin 60°}\)
(b) From an aeroplane in the air and at a horizontal distance of 1050m, the angles of depression of the top and base of a control tower at an instance are 36° and 41° respectively. Calculate, correct to the nearest meter, the :
(i) height of the control tower ; (ii) shortest distance between the aeroplane and the base of the control tower.
Answer Details
(a) Use exact surd values: \(\tan 60° = \sqrt{3}\), \(\cos 30° = \dfrac{\sqrt{3}}{2}\), \(\sin 60° = \dfrac{\sqrt{3}}{2}\).
Numerator: \(2\sqrt{3} + \dfrac{\sqrt{3}}{2} = \dfrac{4\sqrt{3} + \sqrt{3}}{2} = \dfrac{5\sqrt{3}}{2}\).
\(\dfrac{2\tan 60° + \cos 30°}{\sin 60°} = \dfrac{5\sqrt{3}/2}{\sqrt{3}/2} = \mathbf{5}\)
(b) Horizontal distance to the tower \(= 1050\text{ m}\). The top has depression \(36°\) and the base \(41°\). The vertical drop from the aeroplane's level to each point is \(1050\tan(\text{angle})\).
(i) Height of tower \(=\) (drop to base) \(-\) (drop to top):
\(= 1050\tan 41° - 1050\tan 36° = 1050(0.8693 - 0.7265)\)
\(= 1050 \times 0.1428 = 149.9 \approx \mathbf{150\text{ m}}\).
(ii) Shortest distance (line of sight to the base):
\(= \dfrac{1050}{\cos 41°} = \dfrac{1050}{0.7547} = 1391.3 \approx \mathbf{1391\text{ m}}\).
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