IGCSE Maths / Further trigonometry

The ambiguous case of the sine rule

Edexcel International GCSE Maths (4MA1), chapter 25: Further trigonometry, specification point 4.8c. Free lesson.

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The sine rule can give two possible angles because two angles between 0° and 180° can have the same sine. You need to check the wording or diagram to decide which angle fits.

Why one sine value can give two angles

Between 0° and 180°, the sine graph rises to a peak at 90° and then falls in a mirror image. This means an angle and its partner on the other side of 90° have the same sine value. For example, sin 60° ≈ 0.87 and sin 120° ≈ 0.87. In general, sin(180° − x) = sin x.

When two triangles can fit

The ambiguous case can happen when you know two sides and an angle that is not between them. The sine rule may find an acute angle, but a second triangle can have the supplementary angle, which adds to 180° with the first. A calculator gives the acute angle. It cannot tell which of the two possible angles the question intends. The second angle is found by subtracting the acute answer from 180°.

How to check both possible angles

Use the sine rule to find the acute angle first. Then calculate its partner and check whether that angle fits the question and any diagram.

What the question wording tells you

Both angles may satisfy the given side and angle facts. A word such as obtuse, or a diagram showing a wide angle, tells you which triangle to use. The wording is part of the information, not decoration. If the question says the angle is obtuse, choose the answer greater than 90°. If the question does not rule out either triangle, check both possibilities.

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