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Trigonometry

Law of sines

In any triangle, each side divided by the sine of the opposite angle gives the same number.

asin⁡A=bsin⁡B=csin⁡C

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

B₁B₂ACb
  • Height from C: b sin A ≈ 5; side a = 6
  • a swings to two places: two different triangles (the ambiguous case)

What the symbols mean

a
the side opposite angle A
A
the angle opposite side a
b
the side opposite angle B
B
the angle opposite side b
c
the third side, opposite angle C

When to use it

Use it in a triangle without a right angle when you know a side and the angle opposite it, plus one more side or angle: two angles and a side (AAS or ASA), or two sides and an angle that is not between them (SSA). The SSA case can be ambiguous: the known side can swing to meet the base in two places, one, or none, and the calculator reports every triangle that fits. If the known angle is between the two known sides, use the law of cosines instead.

Worked example

In triangle ABC, a = 6, A = 30° and b = 10. Find angle B.

  1. Start from the law of sines and rearrange for sin B.

    asin⁡A=bsin⁡B
  2. sin⁡B=b⁢sin⁡(A)a
    =10⁢sin⁡(30°)6
    =10·0.56
    =56
    ≈0.833333

  3. This is the side-side-angle case, which can be ambiguous: two angles between 0 and 180° have this sine, 56.4427° and 180° − 56.4427° = 123.557°. Each is a valid triangle only if, with the known angle A = 30°, the angles total less than 180°.

  4. B = 56.4427° works: the third angle is 93.5573° and the third side is 11.9769.

  5. B = 123.557° works: the third angle is 26.4427° and the third side is 5.34363.

  6. Two different triangles fit these measurements (the ambiguous case).

Answer B ≈ 56.44° or B ≈ 123.56°: two different triangles fit

Why it works

Drop the height h from C to side c. In the two right triangles that makes, h=bsin⁡A and h=asin⁡B. Setting them equal and dividing gives a/sin⁡A=b/sin⁡B. The common value is the diameter of the circle through the triangle's three corners.

Common mistakes

  • Taking only the calculator's answer for an angle. The inverse sine gives the acute angle; its supplement (180° minus it) may also work.
  • Pairing a side with the wrong angle: each side goes with the angle opposite it, not next to it.
  • Using it when the known angle sits between the two known sides. That is the law of cosines' job.