(a) If \(\cos \alpha = 0.6717\), use mathematical tables to find (i) \(\alpha\) ; (ii) \(\sin \alpha\) (b) The angle of depression of a point P on the groun...
Question 1 Report
(a) If \(\cos \alpha = 0.6717\), use mathematical tables to find (i) \(\alpha\) ; (ii) \(\sin \alpha\)
(b) The angle of depression of a point P on the ground, from the top T of a building is 23.6°. If the distance of P from the foot of the building is 50m, calculate the height of the building, correct to the nearest metre.
Answer Details
(a)(i) \(\cos\alpha = 0.6717\). From the cosine tables, \(\alpha = \cos^{-1}(0.6717) \approx \mathbf{47.8°}\).
(ii) Using \(\sin^2\alpha + \cos^2\alpha = 1\):
\(\sin\alpha = \sqrt{1 - (0.6717)^2} = \sqrt{1 - 0.4512} = \sqrt{0.5488} = \mathbf{0.7408}\)
(Equivalently, \(\sin 47.8° = 0.7408\).)
(b) The angle of depression of \(P\) from the top \(T\) equals the angle of elevation of \(T\) from \(P\) (alternate angles), \(= 23.6°\). The horizontal distance from the foot of the building to \(P\) is \(50\text{ m}\).
\(\tan 23.6° = \dfrac{\text{height}}{50}\)
\(\text{height} = 50 \times \tan 23.6° = 50 \times 0.4369 = 21.85\text{ m}\)
Height of the building \(\approx \mathbf{22\text{ m}}\) (to the nearest metre).
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