Bearings and navigation – journeys, return bearings, angles of elevation and depression
MathematicsTrigonometryAges 15–16
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Sign in to playThree-figure bearings measured clockwise from north, and back bearings (±180°). On the Journey screen a route of 1–3 legs (bearing and distance) is drawn to scale, and the distance and bearing back to the start are found with the cosine rule, the sine rule or east–north components; drag on the map to measure. A sailing challenge and an elevation and depression screen with a lighthouse and boats complete the set.
Lesson: Bearings, solving triangles in navigation problems, angles of elevation and depression
What it shows
A bearing is an angle measured clockwise from north and written with three figures, such as 060° or 245°. Because the north lines at any two points are parallel, the back bearing differs by exactly 180°. A journey of two legs forms a triangle: the cosine rule gives the distance back to the start and the sine rule gives the angle needed for the bearing; with three legs the east and north components are added. Angles of elevation and depression are measured from the horizontal, so tan α = h/d. The model treats the area as flat, ignores wind and currents, and ignores the observer's eye height.
How to use
On Bearing, drag Q around P or use the slider, then read the bearing and the back bearing. On Journey, set the Number of legs and each leg's Bearing and Distance, or drag points B, C and D; drag on empty map to measure. Untick Show answers to practise first. On Sailing challenge, type your Bearing and Distance and press ▶ Sail. On Elevation & depression, change Height h and Distance d₁.
Parameters you can change
- Screen Bearing, Journey, Sailing challenge, Elevation and depression
- Bearing of Q from P (Bearing screen) 0–359 °
- Number of legs in the journey 1–3
- Bearing of leg 1 0–359 °
- Distance of leg 1 1–100 km
- Bearing of leg 2 0–359 °
- Distance of leg 2 1–100 km
- Bearing of leg 3 0–359 °
- Distance of leg 3 1–100 km
- Show answers (Bearing and Journey screens)
- Height h of the top of the lighthouse above sea level 10–300 m
- Distance d₁ from the foot of the lighthouse to the boat 20–1000 m
- Add a second boat
- Distance d₂ from the foot of the lighthouse to the second boat 10–1000 m
Questions to explore
- Why does a back bearing always differ from the bearing by exactly 180°?
- A ship sails 40 km on 060° then 30 km on 140°. On what bearing and how far must it sail to return?
- As a boat approaches the lighthouse, how does the angle of depression from the top change, and why?
