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
The sine rule works for any nondegenerate triangle, right angled or not: 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, plus another known side or angle. Switch to the cosine rule when two sides and the included angle are known, or when all three sides are known.
Finding an angle hides one trap. Rearrange the sine rule to sin B equals b times sin A over a, then take the inverse sine. Sine is shared by supplementary angles, so your calculator hands you the principal value, which is acute or exactly 90 degrees. Always test 180 minus the calculator value too: 30 and 150 share a sine, and 90 is its own supplement. Check both candidates against every given fact: the third angle must stay positive, and any stated angle restrictions must hold. Never pick an answer just because a sketch looks obtuse.
Try it: angle A is 50 degrees with opposite side 7 cm, and angle B is 80 degrees. Pair a with A and b with B, then write b over sin 80 equals 7 over sin 50. Since sin 80 is about 0.985 and sin 50 is about 0.766, b equals 7 times 0.985 over 0.766, which is about 9.0 cm.
When no full side-angle pair exists, stop and check what else is given. A missing angle may come from the 180 degree sum, or the data may be insufficient. And always label the triangle before touching any formula: pairing a side with its neighbour rather than its opposite is a common error to check for.
Pair sides with opposite angles, test the obtuse partner against every given fact, and check the data is sufficient before choosing a rule.
Three readings, six practice questions and six flashcards were written for this sine rule topic. The videos are linked from their original sources.
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Where this leads
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14 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.