Bearings

Simple Bearings We have learnt how to find directions with a compass in primary school. The eight directions To describe the directions of a point relative to another point more precisely, true bearing or compass bearing may be used.

True Bearing When using the true bearing, directions are measured from the north in a clockwise direction. P clockwise direction x° It is expressed as x°, where 0 ≤ x < 360 and the integral part of x must consist of 3 digits.

The true bearings of A, B and C measured from O are For example, A 065° 65° 120° B 273° The true bearings of A, B and C measured from O are , and respectively. 065° 120° 273° The true bearing must consist of 3 digits.

Compass Bearing When using the compass bearing, directions are measured from the north or the south. It is expressed as Nx°E, Nx°W, Sx°W or Sx°E, where 0 < x < 90. B A C D

For example, The compass bearing of A from O is N26°E, and the compass bearing of B from O is S65°W.

Follow-up question (a) Find the compass bearing of B from A. (b) Find the true bearing of A from B. Solution (a) The compass bearing of B from A is N A B W ° 65 N65°W. (b) The true bearing of A from B is ∠CBA. ∠CBA + ∠BAD = 180° (int. ∠s, CB // DA) ∴ The true bearing of A from B is 115°.

Problems Involving Bearings I walk 4 km due east from A to B, and then walk 6 km due south to C. If I want to return to A by the shortest route, which direction should I take? N N 4 km A B the shortest route 6 km C

With the notations in the figure, the required direction is N °W. 4 km A B tan = BC AB ° 6 km 6 km 4 km = ° ° 33.7 N ) 0.1 nearest the to (cor. 33.7 ° = C So, the direction I should take is N33.7°W.

Follow-up question A ship sails in the direction S36°E from A to B for 50 km. Then it sails in the direction N54°E from B to C for 120 km. (a) What is the distance between A and C? Solution D E F With the notations in the figure, (a) ∵ ∠EBA = ∠DAB (alt. ∠s, EB // AD) = 36° ∴ ∠ABC = ∠EBA + ∠EBC = 36° + 54° = 90°

Follow-up question (cont’d) A ship sails in the direction S36°E from A to B for 50 km. Then it sails in the direction N54°E from B to C for 120 km. (a) What is the distance between A and C? Solution D E F (a) Consider △ABC. 2 + = BC AB AC (Pyth. theorem) 130 km = km 120 50 2 + = ∴ The distance between A and C is 130 km.

Follow-up question (cont’d) A ship sails in the direction S36°E from A to B for 50 km. Then it sails in the direction N54°E from B to C for 120 km. (b) Find the compass bearing of A from C, correct to 3 significant figures. Solution D E F (b) Consider △ABC. tan = Ð BC AB ACB km 120 50 = ∴

Follow-up question (cont’d) A ship sails in the direction S36°E from A to B for 50 km. Then it sails in the direction N54°E from B to C for 120 km. (b) Find the compass bearing of A from C, correct to 3 significant figures. Solution D E F (b) ∵ ∠FCB = ∠EBC (alt. ∠s, CF // EB) = 54° ∴ Ð + = ACB FCB ACF ° + » 61986 . 22 54 (cor. to 3 sig. fig.) ° = 6 . 76 ∴ The compass bearing of A from C is S76.6°W.