The Sine Rule
Use a/sin A = b/sin B in triangles with no right angle to find a missing side or angle, and check whether an obtuse second answer also fits.
What a learner can do afterwards
- Pair each side with the angle opposite it before substituting
- Find a missing angle and test whether the obtuse partner also works
- Choose the sine rule when a side and its opposite angle are both known
1 · Read
Triangles with no right angle defeat Pythagoras, so you need the sine rule: a over sin A equals b over sin B equals c over sin C. Before you substitute anything, pair each side with the angle opposite it. Reach for this rule when a side and its opposite angle are both known.
Finding an angle hides one trap. Sine is shared by supplementary angles, so your calculator hands you the acute one while the diagram may show an obtuse angle. Always test 180 minus the calculator value too: 30 and 150 share a sine, and only the diagram tells you which one fits.
Try it: angle A is 30 degrees with opposite side 4 cm, and angle B is 90 degrees. Pair a with A and b with B, then write b over sin 90 equals 4 over sin 30. Since sin 90 is 1 and sin 30 is 0.5, b equals 4 divided by 0.5, which is 8 cm.
When no full side-angle pair exists, put the pen down and switch to the cosine rule instead. And always label the triangle before touching any formula: most lost marks come from pairing a side with its neighbour rather than its opposite.
Pair sides with opposite angles, test the obtuse partner, and switch rules when no pair exists.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 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.