Sine rule for angles

Objective. Calculate angles and recognise ambiguous cases.

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2. Worked example
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Learn: the key idea

The sine rule for angles finds a missing angle in a triangle when two sides and a non-included angle are known, using sin A/a = sin B/b, but requires care because the ambiguous case can give two possible triangles. Rearranging gives sin B = b sin A/a; if this value is less than 1, both B and (180° − B) may be valid, so check which fits the triangle's angle sum. For side a = 7, side b = 9 and angle A = 35°, sin B = 9 sin35°/7 ≈ 0.737, giving B ≈ 47.5° or B ≈ 132.5°.

Worked example

In triangle PQR, side p = 10 cm, side q = 14 cm, and angle P = 40°. Use the sine rule for angles to find angle Q, considering the ambiguous case.

1 $Write the sine rule: sin Q/q = sin P/p$
2 $Rearrange: sin Q = q sin P/p = 14 \times sin40°/10 \approx 0.900$
3 $Find Q: Q = sin⁻¹(0.900) \approx 64.2° or Q = 180° - 64.2° = 115.8°$
4 $Check both: 40° + 64.2° = 104.2° (valid) and 40° + 115.8° = 155.8° (also valid)$

Answer: Q ≈ 64.2° or Q ≈ 115.8°

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Common mistake. A frequent error is accepting only the acute angle from the calculator and ignoring the obtuse possibility (180° minus that angle), missing the ambiguous case entirely. Always check whether the obtuse alternative also produces a valid triangle angle sum.

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