All subjects · Mathematics · WMA11 (P1)

The sine rule and the ambiguous case

The sine rule links each side of a triangle to the sine of the angle opposite it. It works for any triangle, not just right-angled ones.

The rule

a / sin A = b / sin B = c / sin C, where side a is opposite angle A, and so on.

When to use it

Use the sine rule when you know a side and its opposite angle, plus one more side or angle. Remember the angles add up to 180°.

The ambiguous case

If you know two sides and an angle that is not between them, sin B = k has two answers between 0° and 180°: B and 180° - B. Both are possible only if the obtuse one still leaves room for the given angle (they must add up to less than 180°). Always check.

Worked example

In triangle ABC, A = 30°, a = 5 cm and b = 8 cm. Find the possible values of B.

  1. sin B / 8 = sin 30° / 5, so sin B = 0.8
  2. B = 53.1° or B = 180° - 53.1° = 126.9°
  3. 30° + 126.9° < 180°, so both are possible

Answer: B = 53.1° or B = 126.9°

Key idea: a/sin A = b/sin B = c/sin C. When finding an angle from two sides and a non-included angle, check whether 180° minus your answer also works.

Check yourself

  1. In triangle ABC, A = 40°, B = 65° and a = 10 cm. Find b to 1 decimal place.
    Show answer

    14.1 cm. b = 10 sin 65° / sin 40° = 14.1 cm.

  2. In triangle ABC, A = 40°, B = 65° and a = 10 cm. Find c to 1 decimal place.
    Show answer

    15.0 cm. C = 180° - 40° - 65° = 75°. c = 10 sin 75° / sin 40° = 15.0 cm.

  3. In triangle ABC, a = 7 cm, b = 9 cm and A = 35°. Find the acute value of B to 1 decimal place.
    Show answer

    47.5 °. sin B = 9 sin 35° / 7 = 0.7375, so B = 47.5°.

  4. In the same triangle (a = 7 cm, b = 9 cm, A = 35°), find the obtuse value of B to 1 decimal place.
    Show answer

    132.5 °. B = 180° - 47.5° = 132.5°. This works because 35° + 132.5° < 180°.

  5. In triangle ABC, A = 50°, a = 12 cm and b = 10 cm. How many different triangles are possible?
    Show answer

    1. sin B = 10 sin 50° / 12 gives B = 39.7° or 140.3°. But 50° + 140.3° > 180°, so only one triangle works.

  6. In which situation might the sine rule give two possible triangles?
    • Two sides and a non-included angle are known
    • Two angles and one side are known
    • All three sides are known
    • Two sides and the included angle are known
    Show answer

    Two sides and a non-included angle are known. Only with two sides and an angle not between them can sin B = k give two valid angles.

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Last updated 7 November 2026. Spotted a mistake? Tell us.