Theory question 112
BearingsIn the diagram, three points A, B and C are on the same horizontal ground. B is 15 m from A, on a bearing of 053°. C is 18m from B on a bearing of 161°. A vertical pole with top T is erected at B such that angle ATB = 58°. Calculate, correct to three significant figures, (a) the length of AC; (b) the bearing of C from A; (c) the height of the pole BT.
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(a) The bearing of A from B is 233°, so ∠ABC = 233 − 161 = 72°.
AC2 = 152 + 182 − 2(15)(18) cos 72° = 549 − 166.9 = 382.1, so AC ≈ 19.5 m.
(b) sin ∠BAC = 18 sin 72°/19.55 = 0.8758, so ∠BAC ≈ 61.1°. Bearing of C from A = 53 + 61.1 ≈ 114°.
(c) In right-angled triangle ABT: tan 58° = AB/BT, so BT = 15/tan 58° = 15/1.600 ≈ 9.37 m.
Find the angle at B from the two bearings first, then use the cosine rule.