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AI HL · 12.10

Geometry & Trigonometry

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12.10.6

Bearings

12.10.6

Bearings

Target: I can measure and write a bearing as three figures, clockwise from north. (+4 more)

By the end of this section

Key points

Bearings

  • A bearing is an angle measured clockwise from north. Write it with three figures: 040°, not 40°.
  • Draw a north line at every point where you need a bearing. North lines are parallel.
  • The back-bearing is the bearing plus 180°, or minus 180° if the sum is over 360°. This comes from alternate angles between parallel north lines.

A two-leg journey

  • Draw and label the diagram first. Mark the bearings on it.
  • The angle inside the triangle at the turning point is found from the back-bearing of the first leg and the bearing of the second leg.
  • Two sides and the angle between them: use the cosine rule for the third side.
  • To find a return bearing: sine rule for a triangle angle, then add or subtract it from a bearing you know, then add 180° if you need the bearing the other way. Round to the nearest degree.

Two bearings to one point

  • Two observers a known distance apart each measure the bearing of the same point (a boat, a fire). The two lines of sight and the line between the observers make a triangle.
  • Draw a north line at each observer. Use the bearings and parallel north lines to find the two angles at the observers. The angle at the point is 180° minus both.
  • You know one side and all three angles: use the sine rule for each distance. If the observers are due east–west, the angle at each is 90° minus or plus the bearing, so check it from the sketch.

On the calculator

Angles and inverse sine

  • Check degree mode.
  • Use the inverse sine for the sine rule. Q is the largest angle, so the other angles are acute and the calculator's answer is the one you want.

Screenshot and video to come

Explore: a bearing and its back-bearing

Drag A and B on the map. Each point has a north line. The bearing is the angle clockwise from north, written with three figures.

Bearing of B from A
Bearing of A from B

One square is 1 km. Click a point, then use the arrow keys to move it by half a kilometre.

Drag A or B to change the bearing.

  1. Drag B right round A. When is the bearing of B from A less than 180°? What does the back-bearing do then?
  2. Put B due east of A, then due south, then due west. Write down both bearings each time. What do you notice?
  3. Why do the two bearings always differ by exactly 180°? Look at the two north lines.
  4. Tick the box and put B south-west of A. What does the working do that it did not do for north-east? Why?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Back-bearing

The bearing of B from A is 072°. Find the bearing of A from B.

8 km072°ABNN
Working
  1. Draw north at both points

    North lines at A and B are parallel

    Parallel lines make alternate angles equal.

  2. Alternate angles

    \(72^\circ + 180^\circ = 252^\circ\)

    Turning clockwise from north at B, you pass the line BA after 72° plus a half turn.

  3. Three figures

    252°

    Bearings have three figures and are measured clockwise from north. If the sum is over 360°, subtract 360°.

Video walkthrough coming
How far back?

A boat leaves port P and sails 12 km on a bearing of 040° to Q. It then sails 9 km on a bearing of 135° to R. Find the distance PR.

12 km9 km?040°135°PQRNN
Working
  1. Angle at Q

    \(180^\circ + 040^\circ = 220^\circ\) is the bearing of P from Q. \(220^\circ - 135^\circ = 85^\circ\)

    Back-bearing of PQ first. The angle at Q is between QP and QR.

  2. Cosine rule

    \(PR^2 = 12^2 + 9^2 - 2(12)(9)\cos 85^\circ = 206.2\)

    Two sides and the angle between them.

  3. Distance

    \(PR = 14.4\) km

    Square root, 3 significant figures.

Video walkthrough coming
The bearing back to port

A boat leaves port P and sails 12 km on a bearing of 040° to Q. It then sails 9 km on a bearing of 135° to R. Find the bearing on which it must sail from R to return directly to P. Give the bearing to the nearest degree.

12 km9 km?040°135°PQRNN
Working
  1. The angle at Q and PR

    Angle at Q = 85°, PR = 14.36 km

    From the cosine rule, as before.

  2. Sine rule for P

    \(\dfrac{\sin P}{9} = \dfrac{\sin 85^\circ}{14.36}\), so \(\sin P = 0.6244\) and \(P = 38.64^\circ\)

    Use the angle you want on top. Q is the largest angle, so P is acute.

  3. Bearing of R from P

    \(040^\circ + 38.64^\circ = 78.64^\circ\)

    The boat turned right at Q, so R is clockwise from PQ.

  4. Back-bearing

    \(78.64^\circ + 180^\circ = 258.6^\circ\)

    Bearing of P from R, to the nearest degree: 259°.

Video walkthrough coming
Two bearings to one boat

Two lighthouses A and B are 12 km apart, with B due east of A. A boat S is seen from A on a bearing of 048° and from B on a bearing of 322°. Find the distance of the boat from A.

12 km048°322°ABSNN
Working
  1. Angle at A

    \(90^\circ-48^\circ=42^\circ\)

    AB points due east, 090°. The boat is 42° north of east.

  2. Angle at B

    \(322^\circ-270^\circ=52^\circ\)

    BA points due west, 270°. The boat is 52° north of west.

  3. Angle at S

    \(180^\circ-42^\circ-52^\circ=86^\circ\)

    Angles in a triangle. Now you have the side AB opposite this angle.

  4. Sine rule

    \(\frac{AS}{\sin52^\circ}=\frac{12}{\sin86^\circ}\)

    AS is opposite the angle at B.

  5. Distance

    \(AS=\frac{12\sin52^\circ}{\sin86^\circ}=9.479\) km, so 9.48 km

    Then BS = 8.05 km in the same way.

Video walkthrough coming