Bearing Problems with the Sine and Cosine Rules
Turn a journey described by bearings into a triangle, then solve it with the sine or cosine rule instead of drawing it to scale.
What a learner can do afterwards
- Draw the triangle a two-leg journey makes and mark the angle each bearing contributes
- Choose the sine or cosine rule from what the sketch gives, then compute the return distance
- Convert a computed angle back into a three-figure bearing for the answer
1 · Read
Bearings run clockwise from north with three figures, so 40 degrees becomes 040. A two-leg journey makes a triangle: draw each leg, then mark the angle each bearing contributes at the turning point. The turn comes from subtracting the bearings, so a 060 leg followed by 150 differs by 90.
Let the sketch tell you which rule fits. Two sides with their included angle, the classic SAS layout, calls for the cosine rule. An angle paired with its opposite side calls for the sine rule. For 60 km and 80 km meeting at 90 degrees, the return side is 100 km.
A yacht sails 60 km, turns through 90 degrees, then sails 80 km. The cosine rule on the SAS layout gives the return distance: root of 3600 plus 6400, which is 100 km. Keep full calculator accuracy until the final rounding.
A computed angle is not automatically the bearing. Draw the north line at the return point and translate clockwise, so 40 degrees becomes 040. Sense-check at the end: a north east return should read between 000 and 090.
Sketch the journey triangle, pick the rule the sketch offers, solve the return, then translate angles back into three-figure bearings.
More to read
10.1 Non-right Triangles: Law of Sines
Bearings journeys make triangles: draw each leg to scale, then mark the angle each bearing contributes at the turning point. Bearings run clockwise from north, so the turn angle comes from subtracting them. A 060 leg followed by 150 differs by 90.
Pick the sine rule when you know opposite pairs, an angle plus its opposite side. Write the full formula with the unknown on top, substitute, and solve. Sketch first, for the sketch tells you which rule fits.
10.2 Non-right Triangles: Law of Cosines
Reach for the cosine rule with two sides and their included angle, the classic SAS layout of a two leg journey. Return distance squared = a squared + b squared minus 2ab cos C. For 60 km and 80 km at 90 degrees this gives 100 km.
With three sides known, the rearranged cosine rule finds angles instead. Compute the angle, then convert back to a bearing if the question asks for one. Keep full calculator accuracy until the final rounding.
8.1 Non-right Triangles: Law of Sines
A computed angle is not automatically the bearing. Bearings measure clockwise from north with three figures, so translate: 40 degrees clockwise from north is bearing 040. Draw the north line at the return point to see it.
Check sense at the end: a north east return should read between 000 and 090. If your digits fall outside, recheck the clockwise direction. The three figure habit, 040 not 40, earns the accuracy mark.
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.