Problems with the sine and cosine rules

Exam problems mix the sine rule, the cosine rule and the area formula with bearings and obtuse angles.

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Obtuse angles

sin x = sin(180° − x), so sin 30° = sin 150° = 0.5.
cos of an obtuse angle is negative: cos 120° = −0.5.
If the cosine rule gives a negative cosine, the angle is obtuse. Triangle 4, 5, 8: cos C = (16 + 25 − 64)40 = −0.575, so C = 125.1°.
The sine rule can give two possible angles (x and 180° − x); check which fits.

Area backwards

Area = 12ab sin C can find an angle.
Area 30 cm2 with sides 8 cm and 10 cm: 30 = 40 sin C, so sin C = 0.75 and C = 48.6° (or 131.4° if the angle is obtuse).

Bearings

A ship sails 7 km from A to B on a bearing of 040°, then 5 km from B to C on a bearing of 150°.
At B the bearing back to A is 220°, so angle ABC = 220° − 150° = 70°.
AC2 = 72 + 52 − 2 × 7 × 5 × cos 70°, so AC = 7.08 km.
Worked example

A triangle has sides 4 cm, 5 cm and 8 cm. Find its largest angle.

  1. The largest angle is opposite the longest side, 8 cm.
  2. cos C = (42 + 52 − 82)(2 × 4 × 5) = −0.575
  3. C = 125.1°

Answer: 125.1°

Key idea

Negative cosine means an obtuse angle. sin x = sin(180° − x). Area = ½ab sin C works backwards too. With bearings, find the angle inside the triangle first.

Key words
Sine rule
a/sin A = b/sin B = c/sin C, used in any triangle when you know a side and its opposite angle.
Cosine rule
a^{2} = b^{2} + c^{2} − 2bc cos A, used in any triangle when you know two sides and the angle between them, or all three sides.
Formula
A rule linking two or more variables, such as A = πr^{2}.
Obtuse angle
An angle between 90° and 180°.
Bearing
A direction measured clockwise from north, written with three figures, such as 045°.

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. Work out cos 180°.

    Show the answer

    Answer: -1

    cos 180° = −1.

  2. In triangle PQR, p = 5, q = 7 and r = 9. Which formula finds angle R?

    1. cos R = 52 + 72 − 922 × 5 × 7
    2. tan R = 57
    3. cos R = 52 + 72 + 922 × 5 × 7
    4. sin R9 = sin P5
    Show the answer

    Answer: cos R = 52 + 72 − 922 × 5 × 7

    Three sides known: rearranged cosine rule, with r opposite R subtracted.

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Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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