Bearings with trigonometry

Objective. Construct and solve scale and bearing triangles.

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Learn: the key idea

Bearings with trigonometry combine three-figure bearings (measured clockwise from north, always given as 3 digits) with right-angled triangle trigonometry to find distances or directions. Draw a diagram, mark north lines at each point, identify the right-angled triangle formed by the journey legs, then apply SOH-CAH-TOA or the sine/cosine rule. For example, a ship sails 50 km on a bearing of 060° — resolve this into north and east components using sine and cosine of 60°. Always redraw bearings as a scale-style diagram before calculating, since angles are easy to misplace otherwise.

Worked example

A boat sails 40 km due north then 30 km due east. Using bearings and trigonometry, find the bearing of the boat's final position from its start, to the nearest degree.

1 $Right-angled triangle: north leg 40 km, east leg 30 km$
2 $Bearing angle from north: tan θ = opposite/adjacent = 30/40$
3 $θ = tan⁻¹(0.75) \approx 36.87°$
4 $Bearing from north (clockwise) \approx 037°$

Answer: 037°

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Common mistake. A frequent mistake is measuring the bearing angle from the wrong line; always draw a fresh north arrow at the specific point the bearing is measured from, not from a triangle's edge.

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